In Exercises 117118, express each polynomial in standard form. a. Write a polynomial

To multiply the variables in monomials, retain each variable and ______________ the exponents.

To multiply 7x3 (4x5 - 8x2 + 6), use the ______________ property to multiply each term of the trinomial ______________ by the monomial ______________.

To multiply (5x + 3)(x2 + 8x + 7), begin by multiplying each term of x2 + 8x + 7 by ______________. Then multiply each term of x2 + 8x + 7 by ______________. Then combine ______________ terms.

When using the FOIL method to find (x + 7)(3x + 5), the product of the first terms is ______________, the product of the outside terms is ______________, the product of the inside terms is ______________, and the product of the last terms is ______________.

(A + B) 2 = ______________. The square of a binomial sum is the first term ______________ plus 2 times the ______________ plus the last term ______________.

(A - B) 2 = ______________. The square of a binomial difference is the first term squared ______________ 2 times the ______________ ______________ the last term squared.

(A + B)(A - B) = ______________. The product of the sum and difference of the same two terms is the square of the first term ______________ the square of the second term.

If f(x) = x2 - 4x + 7, we find f(a + h) by replacing each occurrence of ______________ with ______________.

4x2 (3x + 2)

5x2 (6x + 7)

2y(y2 - 5y)

3y(y2 - 4y)

5x3 (2x5 - 4x2 + 9)

6x3 (3x5 - 5x2 + 7)

4xy(7x + 3y)

5xy(8x + 3y)

3ab2 (6a2b3 + 5ab)

5ab2 (10a2b3 + 7ab)

-4x2y(3x4y2 - 7xy3 + 6)

-3x2y(10x2y4 - 2xy3 + 7)

-4xna3x2n - 5xn + 1 2 xb

-10xna4x2n - 3xn + 1 5 xb

(x - 3)(x2 + 2x + 5)

(x + 4)(x2 - 5x + 8)

(x - 1)(x2 + x + 1)

(x - 2)(x2 + 2x + 4)

(a - b)(a2 + ab + b2 )

(a + b)(a2 - ab + b2 )

(x2 + 2x - 1)(x2 + 3x - 4)

(x2 - 2x + 3)(x2 + x + 1)

(x - y)(x2 - 3xy + y2 )

(x - y)(x2 - 4xy + y2 )

(xy + 2)(x2y2 - 2xy + 4)

(xy + 3)(x2y2 - 2xy + 5)

(x + 4)(x + 7)

(x + 5)(x + 8)

(y + 5)(y - 6)

(y + 5)(y - 8)

(5x + 3)(2x + 1)

(4x + 3)(5x + 1)

(3y - 4)(2y - 1)

(5y - 2)(3y - 1)

(3x - 2)(5x - 4)

(2x - 3)(4x - 5)

(x - 3y)(2x + 7y)

(3x - y)(2x + 5y)

(7xy + 1)(2xy - 3)

(3xy - 1)(5xy + 2)

(x - 4)(x2 - 5)

(x - 5)(x2 - 3)

(8x3 + 3)(x2 - 5)

(7x3 + 5)(x2 - 2)

(3xn - yn )(xn + 2yn )

(5xn - yn )(xn + 4yn )

(x + 3) 2

(x + 4) 2

(y - 5) 2

(y - 6) 2

(2x + y) 2

(4x + y) 2

(5x - 3y) 2

(3x - 4y) 2

(2x2 + 3y) 2

(4x2 + 5y) 2

(4xy2 - xy) 2

(5xy2 - xy) 2

(an + 4bn ) 2

(3an - bn ) 2

(x + 4)(x - 4)

(x + 5)(x - 5)

(5x + 3)(5x - 3)

(3x + 2)(3x - 2)

(4x + 7y)(4x - 7y)

(8x + 7y)(8x - 7y)

(y3 + 2)(y3 - 2)

(y3 + 3)(y3 - 3)

(1 - y5 )(1 + y5 )

(2 - y5 )(2 + y5 )

(7xy2 - 10y)(7xy2 + 10y)

(3xy2 - 4y)(3xy2 + 4y)

(5an - 7)(5an + 7)

(10bn - 3)(10bn + 3)

[(2x + 3) + 4y][(2x + 3) - 4y]

[(3x + 2) + 5y][(3x + 2) - 5y]

(x + y + 3)(x + y - 3)

(x + y + 4)(x + y - 4)

(5x + 7y - 2)(5x + 7y + 2)

(7x + 5y - 2)(7x + 5y + 2)

[5y + (2x + 3)][5y - (2x + 3)]

[8y + (3x + 2)][8y - (3x + 2)]

(x + y + 1) 2

(x + y + 2) 2

(x + 1)(x - 1)(x2 + 1)

(x + 2)(x - 2)(x2 + 4)

Let f(x) = x - 2 and g(x) = x + 6. Find each of the following. a. (fg)(x) b. (fg)(-1) c. (fg)(0)

Let f(x) = x - 4 and g(x) = x + 10. Find each of the following. a. (fg)(x) b. (fg)(-1) c. (fg)(0)

Let f(x) = x - 3 and g(x) = x2 + 3x + 9. Find each of the following. a. (fg)(x) b. (fg)(-2) c. (fg)(0)

Let f(x) = x + 3 and g(x) = x2 - 3x + 9. Find each of the following. a. (fg)(x) b. (fg)(-2) c. (fg)(0)

f(x) = x2 - 3x + 7

f(x) = x2 - 4x + 9

f(x) = 3x2 + 2x - 1

f(x) = 4x2 + 5x - 1

(3x + 4y) 2 - (3x - 4y) 2

(5x + 2y) 2 - (5x - 2y) 2

(5x - 7)(3x - 2) - (4x - 5)(6x - 1)

(3x + 5)(2x - 9) - (7x - 2)(x - 1)

(2x + 5)(2x - 5)(4x2 + 25)

(3x + 4)(3x - 4)(9x2 + 16)

(x - 1) 3

(x - 2) 3

(2x - 7) 5 (2x - 7) 3

(5x - 3) 6 (5x - 3) 4

In Exercises 113114, find the area of the large rectangle in two ways: a. Find the sum of the areas of the four smaller rectangles. b. Multiply the length and the width of the large rectangle using the FOIL method. Compare this product with your answer to part (a).

In Exercises 113114, find the area of the large rectangle in two ways: a. Find the sum of the areas of the four smaller rectangles. b. Multiply the length and the width of the large rectangle using the FOIL method. Compare this product with your answer to part (a).

In Exercises 115116, express each polynomial in standard formthat is, in descending powers of x. a. Write a polynomial that represents the area of the large rectangle. b. Write a polynomial that represents the area of the small, unshaded rectangle. c. Write a polynomial that represents the area of the shaded blue region.

In Exercises 115116, express each polynomial in standard formthat is, in descending powers of x. a. Write a polynomial that represents the area of the large rectangle. b. Write a polynomial that represents the area of the small, unshaded rectangle. c. Write a polynomial that represents the area of the shaded blue region.

In Exercises 117118, express each polynomial in standard form. a. Write a polynomial that represents the area of the rectangular base of the open box. b. Write a polynomial that represents the volume of the open box.

In Exercises 117118, express each polynomial in standard form. a. Write a polynomial that represents the area of the rectangular base of the open box. b. Write a polynomial that represents the volume of the open box.

A popular model of carry-on luggage has a length that is 10 inches greater than its depth. Airline regulations require that the sum of the length, width, and depth cannot exceed 40 inches. These conditions, with the assumption that this sum is 40 inches, can be modeled by a function that gives the volume of the luggage, V, in cubic inches, in terms of its depth, x, in inches. V(x) = x  (x+10)  [40-(x+x+10)] V(x) = x(x+10)(30-2x) V =   olume depth length width: 40 (depth + length) a. Perform the multiplications in the formula for V(x) and express the formula in standard form. b. Use the functions formula from part (a) and the Leading Coefficient Test to determine the end behavior of its graph. c. Does the end behavior to the right make this function useful in modeling the volume of carry-on luggage as its depth continues to increase? d. Use the formula from part (a) to find V(10). Describe what this means in practical terms. e. The graph of the function modeling the volume of carry-on luggage is shown below. Identify your answer from part (d) as a point on the graph. x y 5 1 10 5 20 25 1000 2000 3000 1000 2000 25 20 15 10 5 y = V(x) f. Use the graph to describe a realistic domain, x, for the volume function, where x represents the depth of the carry-on luggage. Use interval notation to express this realistic domain.

Before working this exercise, be sure that you have read the Blitzer Bonus on page 336 . The table shows the four combinations of color genes that a YY yellow lab and a BY black lab can pass to their offspring. B Y Y BY YY Y BY YY a. How many combinations result in a yellow lab with two recessive yellow genes? What is the probability of a yellow lab? b. How many combinations result in a black lab with a recessive yellow gene? What is the probability of a black lab? c. Find the product of Y and 1 2 B + 1 2Y. How does this product model the probabilities that you determined in parts (a) and (b)?

Explain how to multiply monomials. Give an example.

Explain how to multiply a monomial and a polynomial that is not a monomial. Give an example.

Explain how to multiply a binomial and a trinomial.

What is the FOIL method and when is it used? Give an example of the method.

Explain how to square a binomial sum. Give an example.

Explain how to square a binomial difference. Give an example.

Explain how to find the product of the sum and difference of two terms. Give an example with your explanation.

How can the graph of function f g be obtained from the graphs of functions f and g?

Explain how to find f(a + h) - f(a) for a given function f.

y1 = (x - 2) 2 y2 = x2 - 4x + 4

y1 = (x - 4)(x2 - 3x + 2) y2 = x3 - 7x2 + 14x - 8

y1 = (x - 1)(x2 + x + 1) y2 = x3 - 1

y1 = (x + 1.5)(x - 1.5) y2 = x2 - 2.25

Graph f(x) = x + 4, g(x) = x - 2, and the product function, f g, in a [-6, 6, 1] by [-10, 10, 1] viewing rectangle. Trace along the curves and show that (f g)(1) = f(1) # g(1).

Knowing the difference between factors and terms is important: In (3x2y) 2 , I can distribute the exponent 2 on each factor, but in (3x2 + y) 2 , I cannot do the same thing on each term.

I used the FOIL method to find the product of x + 5 and x2 + 2x + 1.

Instead of using the formula for the square of a binomial sum, I prefer to write the binomial sum twice and then apply the FOIL method.

Special-product formulas have patterns that make their multiplications quicker than using the FOIL method.

If f is a polynomial function, then f(a + h) - f(a) = f(a) + f(h) - f(a) = f(h).

(x - 5) 2 = x2 - 5x + 25

(x + 1) 2 = x2 + 1

Suppose a square garden has an area represented by 9x2 square feet. If one side is made 7 feet longer and the other side is made 2 feet shorter, then the trinomial that represents the area of the larger garden is 9x2 + 15x - 14 square feet.

Express the area of the plane figure shown as a polynomial in standard form.

In Exercises 144145, represent the volume of each figure as a polynomial in standard form.

In Exercises 144145, represent the volume of each figure as a polynomial in standard form.

Simplify: (yn + 2)(yn - 2) - (yn - 3)2 .

The product of two consecutive odd integers is 22 less than the square of the greater integer. Find the integers.

Solve: 3x + 4 10. (Section 4.3, Example 6)

Solve: 2 - 6x 20. (Section 4.1, Example 2)

Write in scientific notation: 8,034,000,000. (Section 1.7, Example 2)

Replace each boxed question mark with a polynomial that results in the given product. a. 3x3 # ? = 9x5 b. 2x3y2 # ? = 12x5y4

(x - 5)(x2 + 3)

(x + 4)(3x - 2y)