5.4 Factoring Polynomials - Big Ideas Learning

5.4TEXAS ESSENTIALKNOWLEDGE AND SKILLS2A.7.D2A.7.EFactoring PolynomialsEssential QuestionHow can you factor a polynomial?Factoring PolynomialsWork with a partner. Match each polynomial equation with the graph of its relatedpolynomial function. Use the x-intercepts of the graph to write each polynomial infactored form. Explain your reasoning.a. x2 5x 4 0b. x3 2x2 x 2 0c. x3 x2 2x 0d. x3 x 0e. x 4 5x2 4 0f. x 4 2x3 x2 2x 0A.B.4 64 666 4C. 4D.4 64 666 4 4E.USINGPROBLEM-SOLVINGSTRATEGIESTo be proficient in math,you need to check youranswers to problems andcontinually ask yourself,“Does this make sense?”F.4 64 666 4 4Factoring PolynomialsWork with a partner. Use the x-intercepts of the graph of the polynomial functionto write each polynomial in factored form. Explain your reasoning. Check youranswers by multiplying.a. f(x) x2 x 2b. f (x) x3 x2 2xc. f(x) x3 2x2 3xd. f(x) x3 3x2 x 3e. f(x) x 4 2x3 x2 2xf. f(x) x 4 10x2 9Communicate Your Answer3. How can you factor a polynomial?4. What information can you obtain about the graph of a polynomial functionwritten in factored form?Section 5.4Factoring Polynomials231

5.4 LessonWhat You Will LearnFactor polynomials.Use the Factor Theorem.Core VocabulVocabularylarryfactored completely, p. 232factor by grouping, p. 233quadratic form, p. 233Previouszero of a functionsynthetic divisionFactoring PolynomialsPreviously, you factored quadratic polynomials. You can also factor polynomialswith degree greater than 2. Some of these polynomials can be factored completelyusing techniques you have previously learned. A factorable polynomial with integercoefficients is factored completely when it is written as a product of unfactorablepolynomials with integer coefficients.Finding a Common Monomial FactorFactor each polynomial completely.a. x3 4x2 5xb. 3y5 48y3c. 5z4 30z3 45z2SOLUTIONa. x3 4x2 5x x(x2 4x 5)Factor common monomial. x(x 5)(x 1)Factor trinomial.b. 3y5 48y3 3y3(y2 16)Factor common monomial. 3y3(y 4)(y 4)Difference of Two Squares Patternc. 5z4 30z3 45z2 5z2(z2 6z 9) 5z2(z 3)2Monitoring ProgressFactor common monomial.Perfect Square Trinomial PatternHelp in English and Spanish at BigIdeasMath.comFactor the polynomial completely.1. x3 7x2 10x2. 3n7 75n53. 8m5 16m4 8m3In part (b) of Example 1, the special factoring pattern for the difference of two squareswas used to factor the expression completely. There are also factoring patterns that youcan use to factor the sum or difference of two cubes.Core ConceptSpecial Factoring PatternsSum of Two CubesExamplea3 b3 (a b)(a2 ab b2)64x3 1 (4x)3 13 (4x 1)(16x2 4x 1)Difference of Two Cubesa3 b3 (a b)(a2 ab Exampleb2)27x3 8 (3x)3 23 (3x 2)(9x2 6x 4)232Chapter 5Polynomial Functions

Factoring the Sum or Difference of Two CubesFactor (a) x3 125 and (b) 16s5 54s2 completely.SOLUTIONa. x3 125 x3 53Write as a3 b3. (x 5)(x2 5x 25)Difference of Two Cubes Patternb. 16s5 54s2 2s2(8s3 27)Factor common monomial. 2s2 [(2s)3 33]Write 8s3 27 as a3 b3. 2s2(2s 3)(4s2 6s 9)Sum of Two Cubes PatternFor some polynomials, you can factor by grouping pairs of terms that have acommon monomial factor. The pattern for factoring by grouping is shown below.ra rb sa sb r(a b) s(a b) (r s)(a b)Factoring by GroupingFactor (a) x3 3x2 7x 21 and (b) z 4 2z 3 27z 54 completely.SOLUTIONa. x3 3x2 7x 21 x2(x 3) 7(x 3) (x2Factor by grouping. 7)(x 3)Distributive Propertyb. z 4 2z 3 27z 54 z3(z 2) 27(z 2) (z3 27)(z 2) (z 2)(z ANALYZINGMATHEMATICALRELATIONSHIPSThe expression 16x 4 81 isin quadratic form becauseit can be written as u2 81where u 4x2.Factor by grouping.3)(z 2Distributive Property 3z 9)Difference of Two Cubes PatternAn expression of the form au2 bu c, where u is an algebraic expression, is said tobe in quadratic form. The factoring techniques you have studied can sometimes beused to factor such expressions.Factoring Polynomials in Quadratic FormFactor 16x4 81 completely.SOLUTION16x4 81 (4x2)2 92Write as a2 b2. (4x2 9)(4x2 9)Difference of Two Squares Pattern (4x2 9)(2x 3)(2x 3)Difference of Two Squares PatternMonitoring ProgressHelp in English and Spanish at BigIdeasMath.comFactor the polynomial completely.4. a3 275. 6z5 750z26. x3 4x2 x 47. 3y3 y2 9y 38. 16n4 6259. 5w6 25w4 30w2Section 5.4Factoring Polynomials233

The Factor TheoremWhen dividing polynomials in the previous section, the examples had nonzeroremainders. Suppose the remainder is 0 when a polynomial f(x) is divided by x k.Then,f(x)x k0x k— q(x) — q(x) where q(x) is the quotient polynomial. Therefore, f(x) (x k) q(x), so that x kis a factor of f (x). This result is summarized by the Factor Theorem, which is a specialcase of the Remainder Theorem.READINGIn other words, x k is afactor of f (x) if and only ifk is a zero of f.Core ConceptThe Factor TheoremA polynomial f(x) has a factor x k if and only if f (k) 0.STUDY TIPDetermining Whether a Linear Binomial Is a FactorIn part (b), notice thatdirect substitution wouldhave resulted in moredifficult computationsthan synthetic division.Determine whether (a) x 2 is a factor of f (x) x2 2x 4 and (b) x 5 is a factorof f(x) 3x4 15x3 x2 25.SOLUTIONa. Find f(2) by direct substitution.b. Find f( 5) by synthetic division.f(2) 22 2(2) 4 53 4 4 4 43Because f (2) 0, the binomialx 2 is not a factor off(x) x2 2x 4.15 1 1500 10255 2550Because f( 5) 0, the binomialx 5 is a factor off(x) 3x4 15x3 x2 25.Factoring a PolynomialShow that x 3 is a factor of f (x) x4 3x3 x 3. Then factor f(x) completely.SOLUTIONShow that f( 3) 0 by synthetic division.ANOTHER WAY 31Notice that you can factorf (x) by grouping.1f (x) x3(x 3) 1(x 3) (x3 1)(x 3) (x 3)(x 1)(x2 x 1) 1 330 300300 10Because f( 3) 0, you can conclude that x 3 is a factor of f (x) by theFactor Theorem. Use the result to write f(x) as a product of two factors and thenfactor completely.f(x) x 4 3x3 x 3Write original polynomial. (x 3)(x3 1) (x 3)(x 234Chapter 5Polynomial Functions1)(x2Write as a product of two factors. x 1)Difference of Two Cubes Pattern

Because the x-intercepts of the graph of a function are the zeros of the function, youcan use the graph to approximate the zeros. You can check the approximations usingthe Factor Theorem.Real-Life Applicationh(t) 4t3 21t2 9t 34DDuringthe first 5 seconds of a roller coaster ride, theffunction h(t) 4t 3 21t 2 9t 34 represents theheighthh (in feet) of the roller coaster after t seconds.HowH long is the roller coaster at or below groundlevellin the first 5 seconds?80h4015 tSOLUTIONS11. Understand the Problem You are given a function rule that represents theheight of a roller coaster. You are asked to determine how long the roller coasteris at or below ground during the first 5 seconds of the ride.22. Make a Plan Use a graph to estimate the zeros of the function and check usingthe Factor Theorem. Then use the zeros to describe where the graph lies belowthe t-axis.33. Solve the Problem From the graph, two of the zeros appear to be 1 and 2.The third zero is between 4 and 5.Step 1 Determine whether 1 is a zero using synthetic division. 14 21 44 25STUDY TIPYou could also checkthat 2 is a zero usingthe original function,but using the quotientpolynomial helps you findthe remaining factor.93425 3434h( 1) 0, so 1 is a zero of hand t 1 is a factor of h(t).0Step 2 Determine whether 2 is a zero. If 2 is also a zero, then t 2 is a factor ofthe resulting quotient polynomial. Check using synthetic division.24 25348 344 17The remainder is 0, so t 2 is afactor of h(t) and 2 is a zero of h.0So, h(t) (t 1)(t 2)(4t 17). The factor 4t 17 indicates that the zero17between 4 and 5 is —, or 4.25.4The zeros are 1, 2, and 4.25. Only t 2 and t 4.25 occur in the first5 seconds. The graph shows that the roller coaster is at or below ground levelfor 4.25 2 2.25 seconds.4. Look Back Use a table ofvalues to verify the positive zerosand heights between the zeros.Xzerozero.51.2522.753.54.255X 2Monitoring ProgressY133.7520.250-16.88-20.25054negativeHelp in English and Spanish at BigIdeasMath.com10. Determine whether x 4 is a factor of f(x) 2x2 5x 12.11. Show that x 6 is a factor of f (x) x3 5x2 6x. Then factor f (x) completely.12. In Example 7, does your answer change when you first determine whether 2 is azero and then whether 1 is a zero? Justify your answer.Section 5.4Factoring Polynomials235

Exercises5.4Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE The expression 9x4 49 is in form because it can be writtenas u2 49 where u .2. VOCABULARY Explain when you should try factoring a polynomial by grouping.3. WRITING How do you know when a polynomial is factored completely?4. WRITING Explain the Factor Theorem and why it is useful.Monitoring Progress and Modeling with MathematicsIn Exercises 5–12, factor the polynomial completely.(See Example 1.)In Exercises 23–30, factor the polynomial completely.(See Example 3.)5. x3 2x2 24x6. 4k5 100k323. y3 5y2 6y 307. 3p5 192p38. 2m6 24m5 64m425. 3a3 18a2 8a 489. 2q4 9q3 18q210. 3r6 11r5 20r426. 2k3 20k2 5k 5011. 10w10 19w9 6w827. x3 8x2 4x 3212. 18v9 33v8 14v729. q4 5q3 8q 40In Exercises 13–20, factor the polynomial completely.(See Example 2.)24. m3 m2 7m 728. z3 5z2 9z 4530. 64n4 192n3 27n 8113. x3 6414. y3 512In Exercises 31–38, factor the polynomialcompletely. (See Example 4.)15. g3 34316. c3 2731. 49k4 932. 4m4 2517. 3h9 192h618. 9n6 6561n333. c4 9c2 2034. y4 3y2 2819. 16t 7 250t420. 135z11 1080z835. 16z4 8136. 81a4 25637. 3r8 3r5 60r238. 4n12 32n7 48n2ERROR ANALYSIS In Exercises 21 and 22, describe andcorrect the error in factoring the polynomial.21.22. 3x3 27x 3x(x2 9) 3x(x 3)(x 3)In Exercises 39–44, determine whether the binomial is afactor of the polynomial. (See Example 5.)39. f(x) 2x3 5x2 37x 60; x 440. g(x) 3x3 28x2 29x 140; x 741. h(x) 6x5 15x4 9x3; x 3x9 8x3 (x3)3 (2x)3 (x3 2x)[(x3)2 (x3)(2x) (2x)2] (x3 2x)(x6 2x4 4x2)42. g(x) 8x5 58x4 60x3 140; x 643. h(x) 6x4 6x3 84x2 144x; x 444. t(x) 48x4 36x3 138x2 36x; x 2236Chapter 5Polynomial Functions