 P.P.2.1: In the exponential form is the ________ and is the ________.
 P.P.5.1: The set of real numbers for which an algebraic expression is define...
 P.P.3.1: For the polynomial the degree is ________, the leading coefficient ...
 P.P.6.1: An ordered pair of real numbers can be represented in a plane calle...
 P.P.1.1: ________ numbers have infinite nonrepeating decimal representations.
 P.1: In Exercises 1 and 2, determine which numbers in the set are (a) na...
 P.P.4.1: The process of writing a polynomial as a product is called ________.
 P.P.2.2: A convenient way of writing very large or very small numbers is cal...
 P.P.5.2: The quotient of two algebraic expressions is a fractional expressio...
 P.P.3.2: A polynomial with one term is called a ________, while a polynomial...
 P.P.6.2: The point of intersection of the  and axes is the ________, and t...
 P.P.1.2: The point 0 on the real number line is called the ________.
 P.2: In Exercises 1 and 2, determine which numbers in the set are (a) na...
 P.P.4.2: A polynomial is ________ ________ when each of its factors is prime.
 P.P.2.3: One of the two equal factors of a number is called a __________ ___...
 P.P.5.3: Fractional expressions with separate fractions in the numerator, de...
 P.P.3.3: To add or subtract polynomials, add or subtract the ________ ______...
 P.P.6.3: The ________ ________ is a result derived from the Pythagorean Theorem
 P.P.1.3: The distance between the origin and a point representing a real num...
 P.3: In Exercises 3 and 4, plot the two rational numbers on the real num...
 P.P.4.3: A ________ ________ ________ is the square of a binomial, and it ha...
 P.P.2.4: In the radical form the positive integer is the ________ of the rad...
 P.P.5.4: Two algebraic expressions that have the same domain and yield the s...
 P.P.3.4: The letters in FOIL stand for the following. F ________ O ________ ...
 P.P.6.4: Finding the average values of the representative coordinates of the...
 P.P.1.4: A number that can be written as the product of two or more prime nu...
 P.4: In Exercises 3 and 4, plot the two rational numbers on the real num...
 P.P.4.4: When a polynomial has more than three terms, a method of factoring ...
 P.P.2.5: Radical expressions can be combined (added or subtracted) when they...
 P.P.5.5: In Exercises 516, find the domain of the expression. 3x 2 9, 2 4x 7
 P.P.3.5: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.5: Exercises 5 and 6, plot the points in the Cartesian plane. 4, 2, 3,...
 P.P.1.5: The ________ of an algebraic expression are those parts separated b...
 P.5: In Exercises 5 and 6, give a verbal description of the subset of re...
 P.P.4.5: In Exercises 58, factor out the common factor. 2x 2 9z 3 6x
 P.P.2.6: The expressions and are ________ of each other.
 P.P.5.6: In Exercises 516, find the domain of the expression. 6x x > 0
 P.P.3.6: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.6: Exercises 5 and 6, plot the points in the Cartesian plane. 7 1, 0.5...
 P.P.1.6: The ________ ________ states that if then or ab 0, a 0 b 0
 P.6: In Exercises 5 and 6, give a verbal description of the subset of re...
 P.P.4.6: In Exercises 58, factor out the common factor. 3z3 6z 2x 2 9z 3
 P.P.2.7: The process used to create a radicalfree denominator is known as _...
 P.P.5.7: In Exercises 516, find the domain of the expression. 13 x
 P.P.3.7: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.7: In Exercises 7 and 8, find the coordinates of the point. The point ...
 P.P.1.7: In Exercises 710, determine which numbers in the set are (a) natura...
 P.7: In Exercises 7 and 8, find the distance between a and b a 74, b 48
 P.P.4.7: In Exercises 58, factor out the common factor. 3xx 58x 52 4x 3
 P.P.2.8: In the expression denotes the ________ to which the base is raised ...
 P.P.5.8: In Exercises 516, find the domain of the expression. 13 x
 P.P.3.8: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.8: In Exercises 7 and 8, find the coordinates of the point. The point ...
 P.P.1.8: In Exercises 710, determine which numbers in the set are (a) natura...
 P.8: In Exercises 7 and 8, find the distance between a and b a 112, b 6
 P.P.4.8: In Exercises 58, factor out the common factor. x 33xx 58x 52 4x 33z...
 P.P.2.9: In Exercises 914, evaluate each expression. (a) 3.33 32 34
 P.P.5.9: In Exercises 516, find the domain of the expression. x2 1x2 2x 1x
 P.P.3.9: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.9: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.9: In Exercises 710, determine which numbers in the set are (a) natura...
 P.9: In Exercises 912, use absolute value notation to describe the situa...
 P.P.4.9: In Exercises 912, find the greatest common factor such that the rem...
 P.P.2.10: In Exercises 914, evaluate each expression. 32 3303
 P.P.5.10: In Exercises 516, find the domain of the expression. x2 5x 6x2 4x
 P.P.3.10: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.10: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.10: In Exercises 710, determine which numbers in the set are (a) natura...
 P.10: In Exercises 912, use absolute value notation to describe the situa...
 P.P.4.10: In Exercises 912, find the greatest common factor such that the rem...
 P.P.2.11: In Exercises 914, evaluate each expression. 2 23 322 3 5
 P.P.5.11: In Exercises 516, find the domain of the expression. x2 2x 3x2 6x 9x2
 P.P.3.11: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.11: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.11: Exercises 11 and 12, plot the real numbers on the real number line....
 P.11: In Exercises 912, use absolute value notation to describe the situa...
 P.P.4.11: In Exercises 912, find the greatest common factor such that the rem...
 P.P.2.12: In Exercises 914, evaluate each expression. 4843 3 34 34
 P.P.5.12: In Exercises 516, find the domain of the expression. x2 x 12x2 8x 16x2
 P.P.3.12: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.12: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.12: Exercises 11 and 12, plot the real numbers on the real number line....
 P.12: In Exercises 912, use absolute value notation to describe the situa...
 P.P.4.12: In Exercises 912, find the greatest common factor such that the rem...
 P.P.2.13: In Exercises 914, evaluate each expression. 4 32 22 31 4 20 4 32 22
 P.P.5.13: In Exercises 516, find the domain of the expression. 4 x 2
 P.P.3.13: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.13: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.13: Exercises 1316, plot the two real numbers on the real number line. ...
 P.13: In Exercises 1316, evaluate the expression for each value of (If no...
 P.P.4.13: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.14: In Exercises 914, evaluate each expression. 2 31 222
 P.P.5.14: In Exercises 516, find the domain of the expression. 2x 5x
 P.P.3.14: In Exercises 514, (a) write the polynomial in standard form, (b) id...
 P.P.6.14: In Exercises 914, determine the quadrant(s) in which is located so ...
 P.P.1.14: Exercises 1316, plot the two real numbers on the real number line. ...
 P.14: In Exercises 1316, evaluate the expression for each value of (If no...
 P.P.4.14: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.15: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.15: In Exercises 516, find the domain of the expression. 1x 3
 P.P.3.15: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.15: In Exercises 15 and 16, sketch a scatter plot of the data shown in ...
 P.P.1.15: Exercises 1316, plot the two real numbers on the real number line. ...
 P.15: In Exercises 1316, evaluate the expression for each value of (If no...
 P.P.4.15: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.16: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.16: In Exercises 516, find the domain of the expression. 1x 2
 P.P.3.16: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.16: In Exercises 15 and 16, sketch a scatter plot of the data shown in ...
 P.P.1.16: Exercises 1316, plot the two real numbers on the real number line. ...
 P.16: In Exercises 1316, evaluate the expression for each value of (If no...
 P.P.4.16: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.17: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.17: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.17: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.17: In Exercises 1722, find the distance between the points. 2, 6, 3, 6...
 P.P.1.17: Exercises 1724, (a) give a verbal description of the subset of real...
 P.17: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.17: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.18: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.18: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.18: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.18: In Exercises 1722, find the distance between the points. 8, 5, 0, 20x,
 P.P.1.18: Exercises 1724, (a) give a verbal description of the subset of real...
 P.18: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.18: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.19: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.19: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.19: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.19: In Exercises 1722, find the distance between the points. 1, 4, 5, 1...
 P.P.1.19: Exercises 1724, (a) give a verbal description of the subset of real...
 P.19: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.19: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.20: In Exercises 1520, evaluate the expression for the given value of x...
 P.P.5.20: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.20: In Exercises 1520, determine whether the expression is a polynomial...
 P.P.6.20: In Exercises 1722, find the distance between the points. 1, 3, 3, 22,
 P.P.1.20: Exercises 1724, (a) give a verbal description of the subset of real...
 P.20: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.20: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.21: In Exercises 2126, simplify each expression. 5x4x2 5z3x
 P.P.5.21: In Exercises 1730, write the rational expression in simplest form.x...
 P.P.3.21: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.21: In Exercises 1722, find the distance between the points. 2, 4 3, 2,...
 P.P.1.21: Exercises 1724, (a) give a verbal description of the subset of real...
 P.21: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.21: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.22: In Exercises 2126, simplify each expression. 4x30 3x, x 0 25
 P.P.5.22: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.22: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.22: In Exercises 1722, find the distance between the points. 9.5, 2.6, ...
 P.P.1.22: Exercises 1724, (a) give a verbal description of the subset of real...
 P.22: In Exercises 1722, identify the rule of algebra illustrated by the ...
 P.P.4.22: In Exercises 1322, completely factor the difference of two squares....
 P.P.2.23: In Exercises 2126, simplify each expression. 3z4 6y 2
 P.P.5.23: In Exercises 1730, write the rational expression in simplest form.y...
 P.P.3.23: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.23: In Exercises 23 and 24, (a) find the length of each side of the rig...
 P.P.1.23: Exercises 1724, (a) give a verbal description of the subset of real...
 P.23: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.23: In Exercises 2328, factor the perfect square trinomial. x 2 4t 1 2 ...
 P.P.2.24: In Exercises 2126, simplify each expression. 7x 2 x3 12x y39x y
 P.P.5.24: In Exercises 1730, write the rational expression in simplest form.x...
 P.P.3.24: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.24: In Exercises 23 and 24, (a) find the length of each side of the rig...
 P.P.1.24: Exercises 1724, (a) give a verbal description of the subset of real...
 P.24: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.24: In Exercises 2328, factor the perfect square trinomial. 4t x 2 4t 1 2
 P.P.2.25: In Exercises 2126, simplify each expression. 4y33y 4 b2 a2b a 2
 P.P.5.25: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.25: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.25: In Exercises 2528, show that the points form the vertices of the in...
 P.P.1.25: Exercises 2530, use inequality notation and interval notation to de...
 P.25: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.25: In Exercises 2328, factor the perfect square trinomial. 9u 2 108y 8...
 P.P.2.26: In Exercises 2126, simplify each expression. 5x2z63x2y21
 P.P.5.26: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.26: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.26: In Exercises 2528, show that the points form the vertices of the in...
 P.P.1.26: Exercises 2530, use inequality notation and interval notation to de...
 P.26: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.26: In Exercises 2328, factor the perfect square trinomial. 36y 9u 2 10...
 P.P.2.27: In Exercises 2730, rewrite each expression with positive exponents ...
 P.P.5.27: In Exercises 1730, write the rational expression in simplest form.2...
 P.P.3.27: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.27: In Exercises 2528, show that the points form the vertices of the in...
 P.P.1.27: Exercises 2530, use inequality notation and interval notation to de...
 P.27: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.27: In Exercises 2328, factor the perfect square trinomial. z 2 z 14
 P.P.2.28: In Exercises 2730, rewrite each expression with positive exponents ...
 P.P.5.28: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.28: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.28: In Exercises 2528, show that the points form the vertices of the in...
 P.P.1.28: Exercises 2530, use inequality notation and interval notation to de...
 P.28: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.28: In Exercises 2328, factor the perfect square trinomial. 9y2 32 y 116 z
 P.P.2.29: In Exercises 2730, rewrite each expression with positive exponents ...
 P.P.5.29: In Exercises 1730, write the rational expression in simplest form. ...
 P.P.3.29: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.29: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.29: Exercises 2530, use inequality notation and interval notation to de...
 P.29: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.29: In Exercises 2936, factor the sum or difference of cubes. x 3 8
 P.P.2.30: In Exercises 2730, rewrite each expression with positive exponents ...
 P.P.5.30: In Exercises 1730, write the rational expression in simplest form.y...
 P.P.3.30: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.30: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.30: Exercises 2530, use inequality notation and interval notation to de...
 P.30: In Exercises 2330, perform the operation(s). (Write fractional answ...
 P.P.4.30: In Exercises 2936, factor the sum or difference of cubes. 27 x 3
 P.P.2.31: In Exercises 3134, write the number in scientific notation. 10,250.4
 P.P.5.31: . Error Analysis Describe the error.
 P.P.3.31: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.31: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.31: Exercises 3140, evaluate the expression. 100
 P.31: In Exercises 3134, simplify each expression. 3x24x33 5y610y
 P.P.4.31: In Exercises 2936, factor the sum or difference of cubes. y 3 64
 P.P.2.32: In Exercises 3134, write the number in scientific notation. 0.000125
 P.P.5.32: Evaluating a Rational Expression Complete the table. What can you c...
 P.P.3.32: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.32: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.32: Exercises 3140, evaluate the expression. 0
 P.32: In Exercises 3134, simplify each expression. 3a26a35y 36x59x10
 P.P.4.32: In Exercises 2936, factor the sum or difference of cubes. x3 827
 P.P.2.33: In Exercises 3134, write the number in scientific notation. One mic...
 P.P.5.33: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.33: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.33: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.33: Exercises 3140, evaluate the expression. 3 84
 P.33: In Exercises 3134, simplify each expression. 2z3 8y0y2
 P.P.4.33: In Exercises 2936, factor the sum or difference of cubes. 8t 3 8 3 1
 P.P.2.34: In Exercises 3134, write the number in scientific notation. Land ar...
 P.P.5.34: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.34: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.34: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.34: Exercises 3140, evaluate the expression. 4 1
 P.34: In Exercises 3134, simplify each expression. x 2)23 40b 35 75b 32 x
 P.P.4.34: In Exercises 2936, factor the sum or difference of cubes. 27x 8t 3 8
 P.P.2.35: In Exercises 3538, write the number in decimal notation. 3.14 104
 P.P.5.35: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.35: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.35: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.35: Exercises 3140, evaluate the expression. 123
 P.35: In Exercises 3538, rewrite each expression with positive exponents ...
 P.P.4.35: In Exercises 2936, factor the sum or difference of cubes. u3 27v3
 P.P.2.36: In Exercises 3538, write the number in decimal notation. 1.801 105
 P.P.5.36: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.36: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.36: In Exercises 2936, (a) plot the points, (b) find the distance betwe...
 P.P.1.36: Exercises 3140, evaluate the expression. 3 3
 P.36: In Exercises 3538, rewrite each expression with positive exponents ...
 P.P.4.36: In Exercises 2936, factor the sum or difference of cubes. x 23 y3 u3
 P.P.4.37: In Exercises 3746, factor the trinomial. x 2 5x 6 2 x 2x
 P.P.2.37: In Exercises 3538, write the number in decimal notation. Light year...
 P.P.5.37: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.37: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.37: An airplane flies from Naples, Italy, in a straight line to Rome, I...
 P.P.1.37: Exercises 3140, evaluate the expression. 5 5
 P.37: In Exercises 3538, rewrite each expression with positive exponents ...
 P.P.4.38: In Exercises 3746, factor the trinomial. x x 2 5x 6 2
 P.P.2.38: In Exercises 3538, write the number in decimal notation. Width of a...
 P.P.5.38: In Exercises 3338, perform the multiplication or division and simpl...
 P.P.3.38: In Exercises 2138, perform the operation and write the result in st...
 P.P.6.38: Sports A soccer player passes the ball from a point that is 18 yard...
 P.P.1.38: Exercises 3140, evaluate the expression. 335
 P.38: In Exercises 3538, rewrite each expression with positive exponents ...
 P.P.4.39: In Exercises 3746, factor the trinomial. s 2 t 6 2 5s 6x
 P.P.2.39: In Exercises 39 and 40, evaluate each expression without using a ca...
 P.P.5.39: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.39: In Exercises 3944, perform the operation. Add 3x 7x 3 4. 3 2x 2 8
 P.P.6.39: Sales The CocaCola Company had sales of $19,564 million in 2002 an...
 P.P.1.39: Exercises 3140, evaluate the expression. 2 , x 2x 2 ,
 P.39: In Exercises 39 and 40, write the number in scientific notation Sal...
 P.P.4.40: In Exercises 3746, factor the trinomial. t s 2 t 6 2
 P.P.2.40: In Exercises 39 and 40, evaluate each expression without using a ca...
 P.P.5.40: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.40: In Exercises 3944, perform the operation. Add 4x 2x 3 x 6. 5 3x 3 2...
 P.P.6.40: . Earnings per Share The earnings per share for Big Lots, Inc. were...
 P.P.1.40: Exercises 3140, evaluate the expression. x 1 x 1 x > 1 x 1
 P.40: In Exercises 39 and 40, write the number in scientific notation Num...
 P.P.4.41: In Exercises 3746, factor the trinomial. 20 y y 2
 P.P.2.41: In Exercises 41 and 42, evaluate each expression without using a ca...
 P.P.5.41: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.41: In Exercises 3944, perform the operation. Subtract 5x x 3 2 3x 8.
 P.P.6.41: In Exercises 4144, find the coordinates of the vertices of the poly...
 P.P.1.41: In Exercises 4144, place the appropriate symbol or between the two ...
 P.41: In Exercises 41 and 42, write the number in decimal notation. Dista...
 P.P.4.42: In Exercises 3746, factor the trinomial. 24 5z z 2 2
 P.P.2.42: In Exercises 41 and 42, evaluate each expression without using a ca...
 P.P.5.42: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.42: In Exercises 3944, perform the operation. Subtract 0.6t 4 2t 2 t . ...
 P.P.6.42: In Exercises 4144, find the coordinates of the vertices of the poly...
 P.P.1.42: In Exercises 4144, place the appropriate symbol or between the two ...
 P.42: In Exercises 41 and 42, write the number in decimal notation. Ratio...
 P.P.4.43: In Exercises 3746, factor the trinomial. 3x 2 x 1 2 5x 224
 P.P.2.43: In Exercises 43 and 44, use the properties of radicals to simplify ...
 P.P.5.43: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.43: In Exercises 3944, perform the operation. Multiply x 72x 3.0.6
 P.P.6.43: In Exercises 4144, find the coordinates of the vertices of the poly...
 P.P.1.43: In Exercises 4144, place the appropriate symbol or between the two ...
 P.43: In Exercises 4348, simplify each radical expression. 3 3 272
 P.P.4.44: In Exercises 3746, factor the trinomial. 2x 3x 2 x 1 2
 P.P.2.44: In Exercises 43 and 44, use the properties of radicals to simplify ...
 P.P.5.44: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.44: In Exercises 3944, perform the operation. Multiply x x 2 2x 1. 2 x ...
 P.P.6.44: In Exercises 4144, find the coordinates of the vertices of the poly...
 P.P.1.44: In Exercises 4144, place the appropriate symbol or between the two ...
 P.44: In Exercises 4348, simplify each radical expression. 3 64125 81
 P.P.4.45: In Exercises 3746, factor the trinomial. 5x 2 3z 2 2 26x 52x
 P.P.2.45: In Exercises 4554, simplify each radical expression. 3 20 128
 P.P.5.45: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.45: In Exercises 4576, multiply or find the special product. x 3x 4x 5
 P.P.6.45: Use the graph below, which shows the minimum wages in the United St...
 P.P.1.45: In Exercises 4550, find the distance between a and b. a 126, b 75
 P.45: In Exercises 4348, simplify each radical expression. 3 216 4 324
 P.P.4.46: In Exercises 3746, factor the trinomial. 9z 5x 2 3z 2 2
 P.P.2.46: In Exercises 4554, simplify each radical expression. 3 1627 75
 P.P.5.46: In Exercises 3946, perform the addition or subtraction and simplify...
 P.P.3.46: In Exercises 4576, multiply or find the special product. x 5x 10
 P.P.6.46: Data Analysis: Exam Scores The table shows the mathematics entrance...
 P.P.1.46: In Exercises 4550, find the distance between a and b. a 126, b 75
 P.46: In Exercises 4348, simplify each radical expression. 2x327 5 64x6
 P.47: In Exercises 4348, simplify each radical expression. 50 18 32 372
 P.P.4.47: In Exercises 4752, factor by grouping. x 2 5x 25 3 x 2 2x 2 9z
 P.P.2.47: In Exercises 4554, simplify each radical expression. 54xy 4 72x3
 P.P.5.47: Error Analysis In Exercises 47 and 48, describe the error
 P.P.3.47: In Exercises 4576, multiply or find the special product. 3x 52x 17x
 P.P.6.47: Using the Midpoint Formula A line segment has as one endpoint and a...
 P.P.1.47: In Exercises 4550, find the distance between a and b. a 452, b 0
 P.48: In Exercises 4348, simplify each radical expression. 3 8x3 2x 18x5 ...
 P.P.4.48: In Exercises 4752, factor by grouping. x 3 5x x 2 5x 25 3
 P.P.2.48: In Exercises 4554, simplify each radical expression. 2182z3
 P.P.5.48: Error Analysis In Exercises 47 and 48, describe the error
 P.P.3.48: In Exercises 4576, multiply or find the special product. 7x 24x 3
 P.P.6.48: Using the Midpoint Formula Use the result of Exercise 47 to find th...
 P.P.1.48: In Exercises 4550, find the distance between a and b. a 14, b 11a 4
 P.P.1.49: In Exercises 4550, find the distance between a and b. a a 9.34, b 5...
 P.49: Explain why 5u 3u 22u.
 P.P.4.49: In Exercises 4752, factor by grouping. 2x 3 x 2 6x 3x
 P.P.2.49: In Exercises 4554, simplify each radical expression. 75x2y
 P.P.5.49: In Exercises 4954, simplify the complex fraction.x2 1x 2
 P.P.3.49: In Exercises 4576, multiply or find the special product. x 2 x 1x 2...
 P.P.6.49: Using the Midpoint Formula Use the Midpoint Formula three times to ...
 P.P.1.50: In Exercises 4550, find the distance between a and b. a 9.34, b 5.65 1
 P.50: The rectangular cross section of a wooden beam cut from a log of di...
 P.P.4.50: In Exercises 4752, factor by grouping. 6 2x 3x3 x4 2x
 P.P.2.50: In Exercises 4554, simplify each radical expression. 5 160x8z 4 4 3...
 P.P.5.50: In Exercises 4954, simplify the complex fraction. x 4x4 4x
 P.P.3.50: In Exercises 4576, multiply or find the special product. 2x2 x 4x2 ...
 P.P.6.50: Using the Midpoint Formula Use the result of Exercise 49 to find th...
 P.P.1.51: In Exercises 5154, use absolute value notation to describe the situ...
 P.51: In Exercises 5154, rationalize the denominator of the expression. T...
 P.P.4.51: In Exercises 4752, factor by grouping. x 5 2x 3 x 2 2 3
 P.P.2.51: In Exercises 4554, simplify each radical expression. 3 1032 618 54
 P.P.5.51: In Exercises 4954, simplify the complex fraction. x2x 12xx 13x 4
 P.P.3.51: In Exercises 4576, multiply or find the special product. x 10x 102x
 P.P.6.51: Make a Conjecture Plot the points and on a rectangular coordinate s...
 P.P.1.52: In Exercises 5154, use absolute value notation to describe the situ...
 P.52: In Exercises 5154, rationalize the denominator of the expression. T...
 P.P.4.52: In Exercises 4752, factor by grouping. 8x5 6x2 12x x 5 2x 3 x 2 2 3...
 P.P.2.52: In Exercises 4554, simplify each radical expression. 5x 3x 29y 10y
 P.P.5.52: In Exercises 4954, simplify the complex fraction. x2 1x x 12x
 P.P.3.52: In Exercises 4576, multiply or find the special product. 2x 32x 3
 P.P.6.52: Collinear Points Three or more points are collinear when they all l...
 P.P.1.53: In Exercises 5154, use absolute value notation to describe the situ...
 P.53: In Exercises 5154, rationalize the denominator of the expression. T...
 P.P.4.53: In Exercises 5356, factor the trinomial by grouping. 2x 2 x 2 2 9x 98x
 P.P.2.53: In Exercises 4554, simplify each radical expression. 348x2 775x2 78...
 P.P.5.53: In Exercises 4954, simplify the complex fraction. x 12xx
 P.P.3.53: In Exercises 4576, multiply or find the special product. x 2yx 2y4
 P.P.6.53: Think About It When plotting points on the rectangular coordinate s...
 P.P.1.54: In Exercises 5154, use absolute value notation to describe the situ...
 P.54: In Exercises 5154, rationalize the denominator of the expression. T...
 P.P.4.54: In Exercises 5356, factor the trinomial by grouping. 6x 2x 2 x 2 2
 P.P.2.54: In Exercises 4554, simplify each radical expression. x 3 7 5x3 7 7 ...
 P.P.5.54: In Exercises 4954, simplify the complex fraction. t 2t 2 1 t 2 1t 2x
 P.P.3.54: In Exercises 4576, multiply or find the special product. 4a 5b4a 5b
 P.P.6.54: Think About It What is the coordinate of any point on the axis? W...
 P.P.1.55: In Exercises 5558, use the bar graph, which shows the receipts of t...
 P.55: In Exercises 55 and 56, rationalize the numerator of the expression...
 P.P.4.55: In Exercises 5356, factor the trinomial by grouping. 6x 2 13x 1 2 x...
 P.P.2.55: In Exercises 5558, rationalize the denominator of the expression. T...
 P.P.5.55: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.55: In Exercises 4576, multiply or find the special product. 2x 32
 P.P.6.55: In Exercises 5557, determine whether the statement is true or false...
 P.P.1.56: In Exercises 5558, use the bar graph, which shows the receipts of t...
 P.56: In Exercises 55 and 56, rationalize the numerator of the expression...
 P.P.4.56: In Exercises 5356, factor the trinomial by grouping. 12x 6x 2 13x 1 2
 P.P.2.56: In Exercises 5558, rationalize the denominator of the expression. T...
 P.P.5.56: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.56: In Exercises 4576, multiply or find the special product. 5 8x2
 P.P.6.56: In Exercises 5557, determine whether the statement is true or false...
 P.P.1.57: In Exercises 5558, use the bar graph, which shows the receipts of t...
 P.57: In Exercises 5760, simplify the expression. 1632
 P.P.4.57: In Exercises 5784, completely factor the expression. 6x 2 48 2 54
 P.P.2.57: In Exercises 5558, rationalize the denominator of the expression. T...
 P.P.5.57: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.57: In Exercises 4576, multiply or find the special product. 4x3 32
 P.P.6.57: In Exercises 5557, determine whether the statement is true or false...
 P.P.1.58: In Exercises 5558, use the bar graph, which shows the receipts of t...
 P.58: In Exercises 5760, simplify the expression. 6423
 P.P.4.58: In Exercises 5784, completely factor the expression. 12x 6x 2 48 2
 P.P.2.58: In Exercises 5558, rationalize the denominator of the expression. T...
 P.P.5.58: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.58: In Exercises 4576, multiply or find the special product. 8x 32 4
 P.P.6.58: HOW DO YOU SEE IT? Use the plot of the point in the figure. Match t...
 P.P.1.59: In Exercises 5962, identify the terms. Then identify the coefficien...
 P.59: In Exercises 5760, simplify the expression. 3x252x1264
 P.P.4.59: In Exercises 5784, completely factor the expression. x 3 16x 3 x2
 P.P.2.59: In Exercises 5962, rationalize the numerator of the expression. The...
 P.P.5.59: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.59: In Exercises 4576, multiply or find the special product. x 13
 P.P.6.59: Proof Prove that the diagonals of the parallelogram in the figure i...
 P.P.1.60: In Exercises 5962, identify the terms. Then identify the coefficien...
 P.60: In Exercises 5760, simplify the expression.x 113x 114 3x2
 P.P.4.60: In Exercises 5784, completely factor the expression. x x 3 16x
 P.P.2.60: In Exercises 5962, rationalize the numerator of the expression. The...
 P.P.5.60: In Exercises 5560, factor the expression by removing the common fac...
 P.P.3.60: In Exercises 4576, multiply or find the special product. x 23
 P.P.1.61: In Exercises 5962, identify the terms. Then identify the coefficien...
 P.61: In Exercises 6164, write the polynomial in standard form. Identify ...
 P.P.4.61: In Exercises 5784, completely factor the expression. x 2 2x 1x
 P.P.2.61: In Exercises 5962, rationalize the numerator of the expression. The...
 P.P.5.61: In Exercises 61 and 62, simplify the expression. 3x13 x233x234
 P.P.3.61: In Exercises 4576, multiply or find the special product. 2x y3
 P.P.1.62: In Exercises 5962, identify the terms. Then identify the coefficien...
 P.62: In Exercises 6164, write the polynomial in standard form. Identify ...
 P.P.4.62: In Exercises 5784, completely factor the expression. 16 6x x 2 x
 P.P.2.62: In Exercises 5962, rationalize the numerator of the expression. The...
 P.P.5.62: In Exercises 61 and 62, simplify the expression. x 31 x 212 2x1 x 2...
 P.P.3.62: In Exercises 4576, multiply or find the special product. 3x 2y3
 P.P.1.63: In Exercises 6366, evaluate the expression for each value of (If no...
 P.63: In Exercises 6164, write the polynomial in standard form. Identify ...
 P.P.4.63: In Exercises 5784, completely factor the expression. 4x 4x 2 6x 1 216
 P.P.2.63: In Exercises 6368, fill in the missing form of the expression. 3 64
 P.P.5.63: In Exercises 6366, simplify the difference quotient. 1x h 1xh
 P.P.3.63: In Exercises 4576, multiply or find the special product. m 3nm 3n3
 P.P.1.64: In Exercises 6366, evaluate the expression for each value of (If no...
 P.64: In Exercises 6164, write the polynomial in standard form. Identify ...
 P.P.4.64: In Exercises 5784, completely factor the expression. 9x 1 4x 4x 2 6...
 P.P.2.64: In Exercises 6368, fill in the missing form of the expression. 2ab34
 P.P.5.64: In Exercises 6366, simplify the difference quotient. 1x h2 1x 2h
 P.P.3.64: In Exercises 4576, multiply or find the special product. x 3yzx 3yz
 P.P.1.65: In Exercises 6366, evaluate the expression for each value of (If no...
 P.65: In Exercises 6568, perform the operation and write the result in st...
 P.P.4.65: In Exercises 5784, completely factor the expression. 2x 2 4x 2x 3
 P.P.2.65: In Exercises 6368, fill in the missing form of the expression. 3x 23
 P.P.5.65: In Exercises 6366, simplify the difference quotient. 1x h 4 1x 4h1
 P.P.3.65: In Exercises 4576, multiply or find the special product. x 3y2
 P.P.1.66: In Exercises 6366, evaluate the expression for each value of (If no...
 P.66: In Exercises 6568, perform the operation and write the result in st...
 P.P.4.66: In Exercises 5784, completely factor the expression. 13x 6 5x 2 2x
 P.P.2.66: In Exercises 6368, fill in the missing form of the expression. 2
 P.P.5.66: In Exercises 6366, simplify the difference quotient. x hx h 1 xx 1h1
 P.P.3.66: In Exercises 4576, multiply or find the special product. x 1y2
 P.P.1.67: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.67: In Exercises 6568, perform the operation and write the result in st...
 P.P.4.67: In Exercises 5784, completely factor the expression. 8x2 196x 116
 P.P.2.67: In Exercises 6368, fill in the missing form of the expression. xx23xxy
 P.P.5.67: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.67: In Exercises 4576, multiply or find the special product. 15x 315x 3
 P.P.1.68: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.68: In Exercises 6568, perform the operation and write the result in st...
 P.P.4.68: In Exercises 5784, completely factor the expression. t 12 49 1
 P.P.2.68: In Exercises 6368, fill in the missing form of the expression. a0.4
 P.P.5.68: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.68: In Exercises 4576, multiply or find the special product. 3x 163x 16
 P.P.1.69: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.69: In Exercises 69 and 70, perform the operation. Add 4x 2x 2 7x 2. 3 ...
 P.P.4.69: In Exercises 5784, completely factor the expression. x 2 12 4x 2t
 P.P.2.69: In Exercises 6978, simplify each expression. 4 323
 P.P.5.69: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.69: In Exercises 4576, multiply or find the special product. 1.5x 41.5x 4
 P.P.1.70: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.70: In Exercises 69 and 70, perform the operation. Subtract 6x4 20x 9x ...
 P.P.4.70: In Exercises 5784, completely factor the expression. x 2 82 36x 2 x 2
 P.P.2.70: In Exercises 6978, simplify each expression. 2 1003
 P.P.5.70: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.70: In Exercises 4576, multiply or find the special product. 3a3 4b23a3...
 P.P.1.71: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.71: In Exercises 7176, find the product. 3x 65x 12 2
 P.P.4.71: In Exercises 5784, completely factor the expression. 5 x 5x 2 x 3x
 P.P.2.71: In Exercises 6978, simplify each expression. 2x232 212 x4 x4y21 3xy...
 P.P.5.71: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.71: In Exercises 4576, multiply or find the special product. 14 x 52
 P.P.1.72: In Exercises 6772, identify the rule(s) of algebra illustrated by t...
 P.72: In Exercises 7176, find the product. x 2x 3x 65x 12 26x4
 P.P.4.72: In Exercises 5784, completely factor the expression. 3u 2u2 6 u3 5
 P.P.2.72: In Exercises 6978, simplify each expression. x3 x12 x3 2 x 51 2 5x5...
 P.P.5.72: In Exercises 6772, rewrite the difference quotient by rationalizing...
 P.P.3.72: In Exercises 4576, multiply or find the special product. 2.4x 32
 P.P.1.73: In Exercises 7376, perform the operation(s). (Write fractional answ...
 P.73: In Exercises 7176, find the product. 2x 36x 56x 52x
 P.P.4.73: In Exercises 5784, completely factor the expression. 2x 2 x 5 3 x2 ...
 P.P.2.73: In Exercises 6978, simplify each expression. 6 x 1
 P.P.5.73: After placing food (at room temperature) in a refrigerator, the tim...
 P.P.3.73: In Exercises 4576, multiply or find the special product. 5xx 13xx 12
 P.P.1.74: In Exercises 7376, perform the operation(s). (Write fractional answ...
 P.74: In Exercises 7176, find the product. 6x 56x 52
 P.P.4.74: In Exercises 5784, completely factor the expression. 5 x3 x 2x 2 x ...
 P.P.2.74: In Exercises 6978, simplify each expression. 6 4 x3
 P.P.5.74: Rate A digital copier copies in color at a rate of 50 pages per min...
 P.P.3.74: In Exercises 4576, multiply or find the special product. 2x 1x 33x ...
 P.P.1.75: In Exercises 7376, perform the operation(s). (Write fractional answ...
 P.75: In Exercises 7176, find the product. 35 2x35 2x
 P.P.4.75: In Exercises 5784, completely factor the expression. 2t 3 16
 P.P.2.75: In Exercises 6978, simplify each expression. 4 32 2x
 P.P.5.75: In Exercises 75 and 76, consider an experiment in which a marble is...
 P.P.3.75: In Exercises 4576, multiply or find the special product. u 2u 2u2 42x
 P.P.1.76: In Exercises 7376, perform the operation(s). (Write fractional answ...
 P.76: In Exercises 7176, find the product. x 43 3
 P.P.4.76: In Exercises 5784, completely factor the expression. 4x2x 12x 122t 3
 P.P.2.76: In Exercises 6978, simplify each expression. 3 10a7 243x 1b
 P.P.5.76: In Exercises 75 and 76, consider an experiment in which a marble is...
 P.P.3.76: In Exercises 4576, multiply or find the special product. x yx yx 2 y 2
 P.P.1.77: Determining the Sign of an Expression Use the real numbers and show...
 P.77: Compound Interest After 2 years, an investment of $2500 compounded ...
 P.P.4.77: In Exercises 5784, completely factor the expression. 53 4x2 83 4x5x...
 P.P.2.77: In Exercises 6978, simplify each expression. x 113x 12 3 x 11 3x 14...
 P.P.5.77: Interactive Money Management The table shows the numbers of U.S. ho...
 P.P.3.77: 7780, find the product. (The expressions are not polynomials, but t...
 P.P.1.78: HOW DO YOU SEE IT? Match each description with its graph. Which typ...
 P.78: Surface Area The surface area of a right circular cylinder is (a) D...
 P.P.4.78: In Exercises 5784, completely factor the expression. 2x 1x 32 3x 12...
 P.P.2.78: In Exercises 6978, simplify each expression. 4x 352 4x 35 3 4x 3524...
 P.P.5.78: Finance The formula that approximates the annual interest rate of a...
 P.P.3.78: 7780, find the product. (The expressions are not polynomials, but t...
 P.P.1.79: In Exercises 79 and 80, determine whether the statement is true or ...
 P.79: Geometry Find a polynomial that represents the total number of squa...
 P.P.4.79: In Exercises 5784, completely factor the expression. 73x 221 x2 3x ...
 P.P.2.79: A funnel is filled with water to a height of centimeters. The formu...
 P.P.5.79: Electrical Engineering The total resistance (in ohms) of two resist...
 P.P.3.79: 7780, find the product. (The expressions are not polynomials, but t...
 P.P.1.80: In Exercises 79 and 80, determine whether the statement is true or ...
 P.80: Geometry Use the area model to write two different expressions for ...
 P.P.4.80: In Exercises 5784, completely factor the expression. x22 x2 14 x 2 ...
 P.P.2.80: HOW DO YOU SEE IT? Package A is a cube with a volume of 500 cubic i...
 P.P.5.80: The mathematical model gives the percent of the normal level of oxy...
 P.P.3.80: 7780, find the product. (The expressions are not polynomials, but t...
 P.P.1.81: Conjecture (a) Use a calculator to complete the table. (b) Use the ...
 P.81: In Exercises 8190, completely factor the expression. x xx 34x 33 xP.4
 P.P.4.81: In Exercises 5784, completely factor the expression. x442x 132x2x 1...
 P.P.2.81: In Exercises 8184, determine whether the statement is true or false...
 P.P.5.81: In Exercises 81 and 82, determine whether the statement is true or ...
 P.P.3.81: Cost, Revenue, and Profit An electronics manufacturer can produce a...
 P.82: In Exercises 8190, completely factor the expression. x xx 34x 33 xP.4
 P.P.4.82: In Exercises 5784, completely factor the expression. x33x2 122xx2 1...
 P.P.2.82: In Exercises 8184, determine whether the statement is true or false...
 P.P.5.82: In Exercises 81 and 82, determine whether the statement is true or ...
 P.P.3.82: Compound Interest After 2 years, an investment of $500 compounded a...
 P.83: In Exercises 8190, completely factor the expression. 25x 2 12x 36 2 49
 P.P.4.83: In Exercises 5784, completely factor the expression. 2x 5435x 4255x...
 P.P.2.83: In Exercises 8184, determine whether the statement is true or false...
 P.P.3.83: Geometry Find a polynomial that represents the total number of squa...
 P.84: In Exercises 8190, completely factor the expression. x 25x 2 12x 36...
 P.P.4.84: In Exercises 5784, completely factor the expression. x2 5324x 344x ...
 P.P.2.84: In Exercises 8184, determine whether the statement is true or false...
 P.P.3.84: Geometry Use the area model to write two different expressions for ...
 P.85: In Exercises 8190, completely factor the expression. x 3 27 3 64
 P.P.4.85: In Exercises 8588, draw a geometric factoring model to represent th...
 P.P.3.85: In Exercises 8590, find the area of the shaded region. Write your r...
 P.86: In Exercises 8190, completely factor the expression. 8x x 3 27
 P.P.4.86: In Exercises 8588, draw a geometric factoring model to represent th...
 P.P.3.86: In Exercises 8590, find the area of the shaded region. Write your r...
 P.87: In Exercises 8190, completely factor the expression. 2x 2 14x 8 2 2...
 P.P.4.87: In Exercises 8588, draw a geometric factoring model to represent th...
 P.P.3.87: In Exercises 8590, find the area of the shaded region. Write your r...
 P.88: In Exercises 8190, completely factor the expression. 3x 2x 2 14x 8
 P.P.4.88: In Exercises 8588, draw a geometric factoring model to represent th...
 P.P.3.88: In Exercises 8590, find the area of the shaded region. Write your r...
 P.89: In Exercises 8190, completely factor the expression. x 2 2x 8 3 x 2...
 P.P.4.89: In Exercises 89 and 90, write an expression in factored form for th...
 P.P.3.89: In Exercises 8590, find the area of the shaded region. Write your r...
 P.90: In Exercises 8190, completely factor the expression. x 3 4x x 2 2x 8
 P.P.4.90: In Exercises 89 and 90, write an expression in factored form for th...
 P.P.3.90: In Exercises 8590, find the area of the shaded region. Write your r...
 P.91: In Exercises 91 and 92, find the domain of the expression. 1x 6
 P.P.4.91: Geometry The cylindrical shell shown in the figure has a volume of ...
 P.P.3.91: Volume of a Box A takeout fastfood restaurant is constructing an ...
 P.92: In Exercises 91 and 92, find the domain of the expression. x 4
 P.P.4.92: The rate of change of an autocatalytic chemical reaction is where i...
 P.P.3.92: Volume of a Box An overnight shipping company is designing a closed...
 P.93: In Exercises 93 and 94, write the rational expression in simplest f...
 P.P.4.93: In Exercises 93 and 94, find all values of for which the trinomial ...
 P.P.3.93: A oneinchwide steel beam has a uniformly distributed load. When t...
 P.94: In Exercises 93 and 94, write the rational expression in simplest f...
 P.P.4.94: In Exercises 93 and 94, find all values of for which the trinomial ...
 P.P.3.94: The stopping distance of an automobile is the distance traveled dur...
 P.95: In Exercises 9598, perform the indicated operation and simplify. x ...
 P.P.4.95: In Exercises 95 and 96, find two integer values of such that the tr...
 P.P.3.95: In Exercises 95 and 96, determine whether the statement is true or ...
 P.96: In Exercises 9598, perform the indicated operation and simplify. 4x...
 P.P.4.96: In Exercises 95 and 96, find two integer values of such that the tr...
 P.P.3.96: In Exercises 95 and 96, determine whether the statement is true or ...
 P.97: In Exercises 9598, perform the indicated operation and simplify. 1x...
 P.P.4.97: In Exercises 97 and 98, determine whether the statement is true or ...
 P.P.3.97: Degree of a Product Find the degree of the product of two polynomia...
 P.98: In Exercises 9598, perform the indicated operation and simplify.3xx...
 P.P.4.98: In Exercises 97 and 98, determine whether the statement is true or ...
 P.P.3.98: Degree of a Sum Find the degree of the sum of two polynomials of de...
 P.99: In Exercises 99 and 100, simplify the complex fraction. 3aa2x1ax 13
 P.P.4.99: 99. Error Analysis Describe the error.
 P.P.3.99: Writing A students homework paper included the following. Write a p...
 P.100: In Exercises 99 and 100, simplify the complex fraction. 12x 3 12x 3...
 P.P.4.100: Think About It Is completely factored? Explain.
 P.P.3.100: HOW DO YOU SEE IT? An open box has a length of inches, a width of i...
 P.101: In Exercises 101 and 102, simplify the difference quotient. 12x h12xh
 P.P.4.101: In Exercises 101 and 102, factor the expression as completely as po...
 P.P.3.101: Think About It Must the sum of two seconddegree polynomials be a s...
 P.102: In Exercises 101 and 102, simplify the difference quotient. 1x h 3 ...
 P.P.4.102: In Exercises 101 and 102, factor the expression as completely as po...
 P.P.3.102: Think About It When the polynomial is subtracted from an unknown po...
 P.103: In Exercises 103 and 104, plot the points in the Cartesian plane. 5...
 P.P.4.103: Think About It Give an example of a polynomial that is prime with r...
 P.P.3.103: Logical Reasoning Verify that is not equal to by letting and and ev...
 P.104: In Exercises 103 and 104, plot the points in the Cartesian plane. 0...
 P.P.4.104: The figure shows a large square with an area of that contains a sma...
 P.105: In Exercises 105 and 106, determine the quadrant(s) in which is loc...
 P.P.4.105: Rewrite as the difference of two squares. Then find a formula for c...
 P.106: In Exercises 105 and 106, determine the quadrant(s) in which is loc...
 P.107: In Exercises 107110, (a) plot the points, (b) find the distance bet...
 P.108: In Exercises 107110, (a) plot the points, (b) find the distance bet...
 P.109: In Exercises 107110, (a) plot the points, (b) find the distance bet...
 P.110: In Exercises 107110, (a) plot the points, (b) find the distance bet...
 P.111: In Exercises 111 and 112, find the coordinates of the vertices of t...
 P.112: In Exercises 111 and 112, find the coordinates of the vertices of t...
 P.113: Sales Barnes & Noble had annual sales of $5.1 billion in 2008 and $...
 P.114: Meteorology The apparent temperature is a measure of relative disco...
 P.115: In Exercises 115 and 116, determine whether the statement is true o...
 P.116: In Exercises 115 and 116, determine whether the statement is true o...
Solutions for Chapter P: Prerequisites
Full solutions for College Algebra  9th Edition
ISBN: 9781133963028
Solutions for Chapter P: Prerequisites
Get Full SolutionsCollege Algebra was written by Patricia and is associated to the ISBN: 9781133963028. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 630 problems in chapter P: Prerequisites have been answered, more than 12200 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter P: Prerequisites includes 630 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
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