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# Beginning Algebra LinC Bridg MAT 0024C

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This 53 page Class Notes was uploaded by Gaetano Shields on Thursday October 29, 2015. The Class Notes belongs to MAT 0024C at Valencia College taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/231209/mat-0024c-valencia-college in Mat Mathematics at Valencia College.

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Date Created: 10/29/15

Notes for Week 12 Name MAT 0024C 72 Multiplying and Dividing Rational Expressions I Multiplying Rational Expressions Multiplying rational expressions Let A B C and D represent polynomials where B and D are not 0 Q B D BD Then simplify ifpossible Examples 3w3x 5yz2 Multi 1 Q py 25yz3 6wx A Q Multiply x 9x x 3 Multi 1 7 Q py y3 W x2 4x 5 5x Multi 1 Q py 5x 5 x5 Dividing Rational Expressions DiViding rational expressions Let A B C and D represent rational polynomials where B C and D are not 0 4 s E B A2 B 39 D B C BC Then simplify ifpossible Examples 2x2 4x2 Q D1V1de 7 2 z z A 2 2 Q Divide x 22x 354x 27x10 3x 27x 6x 12x A Q Divide 5x2l3x 65x2 17x6 x3 x 2 A Unit Conversion Examples 60in1ft 2 1 12m A 288ml 1ft2 lyear 144ml I Q Use the information that follows to determine the miles per uid ounce of gasoline for city and for highway driving for the Chevy Astro Van Hin 1 gallon 128 uid ounces Fuel Type I Regular I MPG m 16 MPG highway 20 73 Adding and Subtracting Rational Expressions with the Same Denominator Adding and Subtracting Rational Expressions Having Like Denominators Adding and Subtracting Rational Expressions If g and g are rational expressions i E A B i E H D D D D D D Then simplify ifpossible Examples Q Subtract 72 A Z 2 Q Subtract b 4 3b 2 3b 2 A 20 2d Q Add CZ d2 62 d2 74 Adding and Subtracting Rational Expressions with Different Denominators Finding the Least Common Denominator Finding the LCD 1 Factor each denominator completely 2 The LCD is a product that uses each different factor obtained in step 1 the greatest number of times it appears in any one factorization Examples Q Findthe LCD of quot1 21 quot11 12m 18m 39 A Q Findthe LCD of b l b1 A 2 Q Findthe LCD of y 10y25 2y 17y35 A Adding and Subtracting Rational Expressions with Unlike Denominators Examples 1 Q Subtract 1 7 9y 4y A 7 Add 77 Q 8b2 6b3 t 2 I Q Add 7 t t3 1 100 b 1b 4b 4239 Q Add Q Subtract 1 2 4 x6 x 8x12 A x Add Q x25x4 x22x1 A Q Subtract y i 32 Denominators that are Opposites Multiplying by 1 When a polynomial is multiplied by l the result is its opposite Examples x Subtract Q x 6 6 x A Q Subtract i t 7 7 t A Notes for Week 5 Name MAT 0024C 64 Factoring Special Products Factoring Perfect Square Trinomials Factoring Perfect Square Trinomials x2 2xyy2 xy2 x2 2xyy2 ye y2 Examples Q Determine whether x2 6x 9 is a perfect square If so factor it Q Determine whether 9x2 26x 16 is a perfect square If so factor it Factor m2 12m36 gt0 Factor x2 24x 144 gt0 Q Factor a2 Zab b2 Factor 25x2 20xy4y2 gt0 Factoring the Difference of Two Squares Factoring a Difference of Two Squares To factor the square of a First quantity minus the square of a Last quantity multiply the First plus the Last by the First minus the Last F2 L2 F LF L Examples Q Factor x2 25 A 0 Factor 22 25 Factor 36x2 121y2 gt0 Factor 12162 144b2 gt0 Factor y4 625 gt0 39 Factor x4 256 O Factoring the sum and difference of cubes Sum ofcubes a3 b3 a ba2 ab b2 Difference of cubes of b3 a ba2 a 4be Examples Q Factor w3 64 A Q Factor x3 1 A Q Factor 1000m3 27713 A 65 Strategies for Factoring A Factoring Strategy Steps for factoring a polynomial l 2 5 Is there a common factor If so factor out the GCF How many terms does the polynomial have If it has two terms look for the following problem types a The difference of two squares b The sum 0ftw0 cubes c The difference of two cubes If it has three terms look for the following problem types a A perfect square trinomial b If the trinomial is not a perfect square use the trialandcheck method or the grouping method If it has four or more terms try to factor by grouping Can any factors be factored further If so factor them completely Does the factorization check Check by multiplying Examples Q Factor 16 402 2522 A Q Factor xzy2 2x2 y2 2 A A Q Factor 70p lq3 35pquotq2 49p5q2 Q Factor 81x4 256y4 A Q Factor 4x2y2 4xy2 y2 A Q Factor m5 y 5y A Q Factor 6x2 x 16 A 66 Solving 39 quot metim bV Factoring I Quadratic Equations 39239 A guadratic eguation is an equation that can be written in the standard form ax2 bx c 0 where a b and 0 represent real numbers and a is not 0 The ZeroFactor Property When the product of two real numbers is 0 at least one ofthem is 0 If a and b represent real numbers and if ab0then a0 0r b0 Examples Q Solve x 8Xx 3 0 A Q Solve 4x2 12x9 0 A Solving Quadratic Equations The factoring method for solving a quadratic equation 1 Write the equation in standard form axz bx c 0 2 Factor the lefthand side 3 Use the zerofactor property 4 Solve each resulting equation 5 Check the results in the original equation Examples Q Solve 5x5x 7 0 A Q Solve 15x2 20x 0 A Q Solve 9y2 l0 A Q Solve 9x2 64 A Q Solve x2 2x 150 A Q Solve 25x2 80x 15 A Q Solve x2x 3 14 A Q Solve x3 22x 9x2 0 A Applications The Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse then a2 b2 c2 Examples Q The formula h l6t2 Vt gives the height h in feet of an object after t seconds when it is shot upward into the air with an initial velocity v in feet per second After how many seconds will the object hit the ground if it is shot with a velocity of 144 feet per second A Q The inclined ramp of the boat launch is 8 meters longer than the rise of the ramp The run is 7 meters longer than the rise How long are the three sides of the ramp Rise Chapter 7 Rational Expressions and Equations 71 Simplif 39ng Rational Expressions I Evaluating Rational Expressions P 39239 A ratlonal express10n1s an express10n of the form 5 where P and Q are polynomials and Q does not equal 0 Examples xZ S Q Find the value of 0 for x 3 x2 A Q Find all values of x for which the rational expression is unde ned A Simplifying Rational Expressions Simplifying rational expressions Factor the numerator and denominator completely to determine their common factors 2 Remove factors equal to 1 by replacing each pair of factors common to the numerator and denominator with the equivalent fraction 1 Examples 4 5 Q Simplify 14x3y 2x y A Q Simplify 2y A 6t 42 Sim lif 7 Q p y t 7 A Q Simplify A Q Simplify A Q Simplify A yz c c2 20 81 y210y939 y2 4y4 23 I d2 dd239 Notes for Week 7 Name MAT 0024C 54 Fvnnnnnf Rules and M n I M I 39 Multiplying Monomials Product rule for exponents To multiply exponential expressions with the same base keep the common base and add the exponents For any number x and any natural numbers 7 and 7 xm 39 xn xmn I Multiplying monomials To multiply two monomials multiply the numerical factors the coe icients and then multiply the variable factors Examples Q Simplify 3234 A Q Multiply 7x3y6 4xy32xy2 A Multiply 47gtlt1010 84gtlt107 and write your answer in simplest form gt10 Q Write an expression in simplest form for the area h k 4h Raising an Exponential Expression to a Power Power rule for exponents To raise an exponential expression to a power keep the base and multiply the exponents For any number x and any natural numbers 7 and 7 W Examples Q Simplify x4 3 A Q Simplify x5x47 A Powers of Products Powers of a Product To raise a product to a power raise each factor of the product to that power For any numbers 7 and 7 and any natural number 7 Wquot x y Examples Q Simplify 4x2 A Q Simplify 3x2y5 3 A 55 39 39 I 39 39 39 Special Products I Multiplying a Polynomial by a Monomial Multiplying polynomials by monomials To multiply a monomial and a polynomial multiply each term of the polynomial by the monomial Q Multiply 5xy 9x3 2x xy 7y3 y A Q Multiply 7cv4 9x3 ch2 x2 8y A Multiplying Binomials Multiplying binomials To multiply two binomials multiply each term of one binomial by each term of the other binomial and then combine like terms The FOIL method First Outside Inside Last Examples Q Multiply 2x 73x 8 A Q Multiply 9x 3x 9 A Q Multiply 8x2 38x2 10 A Q Alarger rectangle is formed out of smaller rectangles a b c 3 1 D Write an expression in simplest form for the length along the top Write an expression in simplest form for the width along the side Write an expression that is the product of the length and width that you found in parts a and b Write an expression in simplest form that is the sum of the areas of each of the smaller rectangles Explain why the expressions in parts c and d are equivalent 5x 6 Multiplying Polynomials Multiplying polynomials To multiply two polynomials multiply each term of one polynomial by each term of the other polynomial and then combine like terms Examples Q Multiply 9xy 2y9x 3x 3y A Q Multiply 6x 8x2 3x 72x2 4x A Special Product Squaring a Binomial Squaring a Binomial The square of a binomial is the square of its rst term plus twice the product of both of its terms plus the square of its second term 2 2 2 2 2 2 xy x 2xyy x y x 2xyy Examples Q Find the square x 32 A Q Findthe square 7x82 A Special Product The Product of a Sum and Difference Multiplying the Sum and Difference of Two Terms The product of the sum and difference of the two terms x and y is the square of x minus the square of y 2 xyxyx2y Examples Q Multiply x 3x 3 A 1 1 Q Multiply 8x gISx A 56 Exponent Rules and Dividing Polynomials Dividing a Monomial by a Monomial Quotient rule for exponents To divide exponential expressions with the same base keep the common base and subtract the exponents For any nonzero number x and any natural numbers 7 and 7 where m gt 7 Examples 9 Q Divide x7 x A yll Q Simplify j y A 24 3 2 Q Divide x y 8xy A 26x5y8 Q D1V1de 3 8 4x y A 9 and write the answer in scientific notation 6 Q Divide 73396X10 24X10 A Dividing a Polynomial by a Monomial Dividing a polynomial by a monomial To divide a polynomial by a monomial divide each term of the polynomial by the monomial Let a b and d represent monomials where d is not 0 ab a b 7ii d d d Examples 8x4 8 12x5 9 Q Divide 4x y A 81y12 32y10 45y6 4 Q Divide 9y A Dividing a Polynomial by a Polynomial Examples x2 x 12 Divide Q 3 A x2 9 Q Divide Q Divide 2x34x2 2x3 x2 A Notes for Week 2 Name MAT 0024C 24 Applying the Principles to Formulas I Isolating a variable Examples Q Solve fora xa3y A Q Solve for w 19 21 2w A Q Solve for a 52na bm c A w x Solveforw 77 Q 4 6 y A Q Solve for v D a v A 25 T 39 quot Word t0 Fanatian Examples Q Translate the sentence to an equation and then solve Six added to p is negative two A Q Translate the sentence to an equation and then solve Three sevenths of a number is equal to negative nineeighths A Q Translate the sentence to an equation and then solve Tripling the difference of a number and ve produces negative fteen Q Translate the sentence to an equation and then solve Five more than four times a number is equal to seven subtracted from that number 26 Solving Linear Inequalities The only difference between solving an inequality and an equality is that if you multiply or divide by a negative number reverse the inequality Symbols Less than lt Less than or equal to S Greater than gt Greater than or equal to 2 Examples Q For x lt 5 1 Write the solution set in setbuilder notation A ii Write the solution set in interval notation A iii Graph the solution set A Q For xZ l 1 Write the solution set in setbuilder notation A 11 Write the solution set in interval notation A iii Graph the solution set A Q For 3y 2 gt10 i Solve ii Write the solution set in setbuilder notation iii Write the solution set in interval notation iV Graph the solution set A Q Solve 7xlt 42 A Q Solve 2x3 217 A Q Solve 3x2l gt 6x 7 A Chapter 5 Polynomials 51 Exponents and Scienti c Notation o 00 0 O An exponent indicates repeated multiplication It tells how many times the m is used as a factor Example 35 33333 Note An exponent only corresponds to what is directly in front of it unless there are parentheses If there are parentheses it corresponds to everything inside the parentheses Examples Q Simplify 53 A Q Simplify 53 A Q Simplify 53 A Powers of Quotients Powers of a Quotient To raise a quotient to a power raise the numerator and the denominator to that power For any numbers m and n and any natural number n where y at 0 Examples 4 2 Q Simplify A 2 3 Q Simplify A Zero Exponents Zero exponents Any nonzero base raised to the 0 power is 1 For any nonzero real number x x l Examples Q Simplify 6 A Q Simplify 3 A Q Simplify 3 70 A Negative Integer Exponents Negative exponents For any nonzero real number x and any integer n x41 n x i and xn xrn In words x7 is the reciprocal of x Examples Q Simplify 7393 A Simplify 6m 2 DgtO Q Simplify 4x712 A 2 72 Q Simplify Converting from Scienti c to Standard Notation 39239 Scienti c notation A positive number is written in scienti c notation when it is written in the form N X10quot Where Is N lt 10 and n is an integer Large numbers greater than 10 have positive exponents Small numbers less than 1 have negative exponents Converting from scienti c to standard notation a If the exponent is positive move the decimal point the same number of places to the right as the exponent b If the exponent is negative move the decimal point the same number of places to the left as the absolute value of the exponent Examples Q Write the number in standard notation 4632 X 106 A Q Write the number in standard notation 83994 X 1039s A Writing Numbers in Scienti c Notation Examples Q Write the number in scienti c notation 843000000 A Q Write the number in scienti c notation 00000000349 Q Write the number in scienti c notation 600340000 A Q Write the number in scienti c notation 0000030540 52 Introduction to Polynomials I Polynomials 39239 A polynomial is a single term or a sum of terms in which all variables have wholenumber exponents No variable appears in the denominator 39239 Polynomials are classi ed according to the number of terms they have A polynomial with exactly one term is called a monomial exactly two terms a binomial and exactly three terms a trinomial Polynomials with four or more 3113 4u2 5t2 4t3 18a2b4ab 27x3 6x 2 29z17 1 a2 2abb2 39239 A m is a product or quotient of numbers andor variables A single number or variable is also a term v The numerical factor of a term is called the coef cient of the term 39239 The degree of a term of a polynomial in one variable is the value of the exponent on the variable If a polynomial is in more than one variable the degree of a term is the sum of the exponents on the variables The degree of a nonzero constant is 0 v The degree of a polynomial is the same as the highest degree of any term of the polynomial Examples Q Describe the polynomial 8x7 6xy2 3 using the following Is it a polynomial Is it a monomial binomial trinomial or does it have no special name What are the terms What are the coef cients of the terms What are the degrees of the terms What is the degree of the polynomial 33833 Evaluating Polynomials Examples Q Evaluate 2x2 7x 3 for x 2 A Q Evaluate 53ny 72 for x 3 y4 and z 1 A Q The polynomial 7r2 2r 6 describes the voltage in a circuit where r represents the resistance in the circuit a Find the voltage if the resistance is 6 ohms b Find the voltage if the resistance is 8 ohms A I Writing Polynomials in Descending Order Examples Q Write x5 x12 6x3 in descending order A Q Write 16y 7y5 22 6y393 in descending order A I Simplifying Polynomials by Combining Like Terms Examples Q Combine like terms 09x4 07x6 02x3 15x6 x3 A Q Combine like terms 34x121 3x13 42x13 x211 4x121 15x13 A 53 Adding and thh m no 1 39 I Adding Polynomials Adding polynomials To add polynomials combine their like terms Examples Q Add 8x7 9x5 2x3 x 9x7 8x5 x A Q Add 14x14 40 y5 7x 8xy 5 y3 8x14 10y5 A Q Write the expression for the perimeter in simplest form 2X3 Subtracting Polynomials Subtracting the polynomials To subtract two polynomials change the signs of the terms of the polynomial being subtracted drop the parentheses and combine like terms Examples Q Subtract 14x5 3x 8 16x5 8x3 9x 2 A Q Subtract 8x2 10x 7 9x2 7x 8 A Q The polynomial 1055m 147571 2750p describes the revenue a pet store generates from the sale of three different litter boxes The expression 573m 826n 1522p describes the cost the store pays to sell each ofthe products Write a polynomial in simplest form that describes the store s net pro t A Notes forWeek7 Name MAT 0024c Chapter 4 Graphing Linear Equa nns anil Inequalities 41 The Rattan lar Cnnn linzte Sgstem The Rectangular Cnnn linzte System A reetangular eoordinate system is formed by two perpendieular number lines The honzontal numberline is usually ealled the x axis and the vertieal numberline is usually ealled the yraxls The point where the axes interseetis ealled the origin The axes form a eoordinate Blane and the dlvlde itinto four regions ealled guadrants whieh are numbered using Roman numerals as shown below iv Eaeh point in a eoordinate plane ean be identified by an ordered gair of real numbers n andy wntten in the form dry The rstnumber n in the pairis ealled the choordlnate andthe seeond numbery is ealled the eoordinate The proeess oflocanng apoint in the eoordinate plane is ealledgraghing or glottingthe point Examples Q P10 each pomt and state the quadrant m whmh xtlxes 3935 a b 00 372 d 015 Q Wm the coordinates ofeach pomt ram Q State the quadrant in which or axis on which 3 7 is located A Q State the quadrant in which or axis on which 0 85 is located A Q State the quadrant in which or axis on which 5 6 is located A Graphing Mathematical Relationships Example Q The table below shows the cost to rent a sailboat for a given number of hours Write the data in the table as ordered pairs and plot them a Is the cost to rent a sailboat linear or nonlinear b What does is cost to rent the boat for 3 hours c Q Determine whether the set of points is linear or nonlinear 1 3 0 0 2 0 3 1 A 42 Graphing Linear Equations Graphing Linear Equations 39239 A linear equation in two variables is an equation that can be written in the form Ax By C where A B and C are real numbers and A and B are not both 0 Graphing Linear Equations Solved for y 1 Find three solutions of the equation by selecting three values for X and calculating the corresponding values of y 2 Plot the solutions on a rectangular coordinate system 3 Draw a straight line passing through the points If the points do not lie on a line check your computations Examples Q Graph y x3 A Q Graph 3y6x 9 Q Graph y 2 Q Graph x3 Q Graph y 2x Applications Q An academic tutor charges 20 for supplies and then 25 per hour of tutoring The equation 0 2571 20 describes the total that he would charge for tutoring where n represents the number of hours of tutoring and c is the total cost i Find the total cost if the tutor works 3 hours ii If a client s total charges are 145 for how many hours of labor was the client charged iii Graph the equation with 71 along the horizontal iV What does the cintercept yintercept represent 43 Graphing Using Intercepts 39239 The point where a graph intersects the XaXis is called the Xintercept Similarly the point where a graph intersects the yaXis is called the yintercept To nd the Xintercept set y0 and solve for X It is in the form i0 To nd the yintercept set X0 and solve for y It is in the form 0i Examples Q Find the X and yintercepts of the equation y 5x3 A Q Find the X and yintercepts of the equation 4y 3x 2 A Q Find the X and yintercepts of the equation y 12x A Q Find the X and yintercepts of the equation x 3 A Q Graph using the X and yintercepts y 3x 6 Q Graph using the X and yintercepts y 4x 2 Q Graph using the X and yintercepts 6y 12 4x 44 Slope Intercept Form SlopeIntercept form y mx b where m slope and b yintercept m B H where xly1 and x2y2 are points on the graph run x2 9 Examples Q For the equation y x 4 determine the slope and the yintercept Graph the equation Q For the equation y 73x 5 determine the slope and the yintercept Graph the equation Q For the equation 2 y 6x 2 determine the slope and the yintercept Graph the equation Q For the equation 3y 9x 6 determine the slope and the yintercept Graph the equation A Q For the equation y 2 3x determine the slope and the yintercept Graph the equation Q Find the slope ofthe line connecting the points 47 and 36 A Q Find the slope ofthe line connecting the points 62 and 37 A Q Find the slope ofthe line connecting the points 07 and 08 A Q Find the slope ofthe line connecting the points 20 and 120 A

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