Chapter 4: Factoring Polynomials

Algebra 2FactoringPolynomials

Algebra 2 Bell RingerMartin-Gay, Developmental Mathematics2

Algebra 2 Bell RingerAnswer:AMartin-Gay, Developmental Mathematics3

Daily Learning Target (DLT)Tuesday February 12, 2013“I can understand, apply, andremember to factor and solvepolynomials By Finding theGreatest Common Factors(GCFs).”Martin-Gay, Developmental Mathematics4

Factoring Polynomials 1Worksheet Assignment1. x56.x 5, x 82. 2x47.x 9, x -73. ab8.x -4/74. a2b29.x 5/3, 7/95. x -15, -2 10. x -5/2, 5/2Martin-Gay, Developmental Mathematics5

§ 13.1The Greatest CommonFactor

Greatest Common FactorGreatest common factor – largest quantity that is afactor of all the integers or polynomials involved.Finding the GCF of a List of Integers or Terms1) Prime factor the numbers.2) Identify common prime factors.3) Take the product of all common prime factors. If there are no common prime factors, GCF is 1.Martin-Gay, Developmental Mathematics7

Factoring PolynomialsThe first step in factoring a polynomial is tofind the GCF of all its terms.Then we write the polynomial as a product byfactoring out the GCF from all the terms.The remaining factors in each term will form apolynomial.Martin-Gay, Developmental Mathematics8

§ 13.3Factoring Trinomials of2the Form ax bx c

Factoring PolynomialsExampleFactor and solve the polynomial 25x2 20x 4 0.Possible factors of 25x2 are {x, 25x} or {5x, 5x}.Possible factors of 4 are {1, 4} or {2, 2}.We need to methodically try each pair of factors until we finda combination that works, or exhaust all of our possible pairsof factors.Keep in mind that, because some of our pairs are not identicalfactors, we may have to exchange some pairs of factors andmake 2 attempts before we can definitely decide a particularpair of factors will not work.Continued.Martin-Gay, Developmental Mathematics10

Factoring PolynomialsExample ContinuedWe will be looking for a combination that gives the sum of theproducts of the outside terms and the inside terms equal to 20x.Factors Factors ResultingProduct ofProduct ofSum ofof 25x2 of 4Binomials Outside Terms Inside Terms Products{x, 25x} {1, 4} (x 1)(25x 4)4x25x29x(x 4)(25x 1)x100x101x{x, 25x} {2, 2} (x 2)(25x 2)2x50x52x{5x, 5x} {2, 2} (5x 2)(5x 2)10x10x20xContinued.Martin-Gay, Developmental Mathematics11

Factoring PolynomialsExample ContinuedCheck the resulting factorization using the FOIL method.FOIL(5x 2)(5x 2) 5x(5x) 5x(2) 2(5x) 2(2) 25x2 10x 10x 4 25x2 20x 4So our final answer when asked to factor 25x2 20x 4will be (5x 2)(5x 2) or (5x 2)2.Martin-Gay, Developmental Mathematics12

Solve For X Now Find The ZerosExampleFactor and solve the polynomial 25x2 20x 4 0.Note, there are other factors, but once we find a pairthat works, we do not have to continue searching.So 25x2 20x 4 (5x 2)2.5x 2 0-2 -25x -255x -2/5Martin-Gay, Developmental Mathematics13

Factoring PolynomialsExampleFactor and solve the polynomial 21x2 – 41x 10 0.Possible factors of 21x2 are {x, 21x} or {3x, 7x}.Since the middle term is negative, possible factors of 10must both be negative: {-1, -10} or {-2, -5}.We need to methodically try each pair of factors untilwe find a combination that works, or exhaust all of ourpossible pairs of factors.Continued.Martin-Gay, Developmental Mathematics14

Factoring PolynomialsExample ContinuedWe will be looking for a combination that gives the sum ofthe products of the outside terms and the inside terms equalto 41x.Factors Factors ResultingProduct ofProduct ofSum ofof 21x2 of 10 Binomials Outside Terms Inside Terms Products–10x 21x– 31x(x – 10)(21x – 1)–x 210x– 211x{x, 21x} {2, 5} (x – 2)(21x – 5)–5x 42x– 47x(x – 5)(21x – 2)–2x 105x{x, 21x}{1, 10}(x – 1)(21x – 10)Martin-Gay, Developmental Mathematics– 107xContinued.15

Factoring PolynomialsExample ContinuedFactors Factors ResultingProduct ofProduct ofSum ofof 21x2 of 10 Binomials Outside Terms Inside Terms Products{3x, 7x}{1, 10}(3x – 1)(7x – 10) 30x 7x 37x(3x – 10)(7x – 1) 3x 70x 73x{3x, 7x} {2, 5} (3x – 2)(7x – 5) 15x 14x 29x(3x – 5)(7x – 2) 6x 35x 41xContinued.Martin-Gay, Developmental Mathematics16

Factoring PolynomialsExample ContinuedCheck the resulting factorization using the FOIL method.FOIL(3x – 5)(7x – 2) 3x(7x) 3x(-2) - 5(7x) - 5(-2) 21x2 – 6x – 35x 10 21x2 – 41x 10So our final answer when asked to factor 21x2 – 41x 10will be (3x – 5)(7x – 2).Martin-Gay, Developmental Mathematics17

Solve For X Now Find The ZerosExampleFactor and solve the polynomial 21x2 – 41x 10 0. Note, there are other factors, but once we find a pairthat works, we do not have to continue searching.So 21x2 – 41x 10 (3x – 5)(7x – 2).3x - 5 0 5 53x 533x 5/37x - 2 0 2 27x 277x 2/7Martin-Gay, Developmental Mathematics18

§ 13.5Factoring Perfect SquareTrinomials and theDifference of Two Squares

Difference of Two SquaresExampleFactor and solve the polynomial x2 – 9 0.The first term is a square and the last term, 9, can bewritten as 32. The signs of each term are different, sowe have the difference of two squaresTherefore x2 – 9 (x – 3)(x 3).Note: You can use FOIL method to verify that thefactorization for the polynomial is accurate.Martin-Gay, Developmental Mathematics20

Solve For X Now Find The ZerosExampleFactor and solve the polynomial x2 – 9 0. Note, there are other factors, but once we find a pairthat works, we do not have to continue searching.So x2 – 9 (x – 3)(x 3).x–3 0 3 3x 3x 3 0- 3 -3x -3Martin-Gay, Developmental Mathematics21

§ 13.4Factoring Trinomials of2the Form x bx cby Grouping

Factoring by GroupingFactoring polynomials often involves additionaltechniques after initially factoring out the GCF.One technique is factoring by grouping.ExampleFactor xy y 2x 2 by grouping.Notice that, although 1 is the GCF for all fourterms of the polynomial, the first 2 terms have aGCF of y and the last 2 terms have a GCF of 2.xy y 2x 2 x · y 1 · y 2 · x 2 · 1 y(x 1) 2(x 1) (x 1)(y 2)Martin-Gay, Developmental Mathematics23

Factoring by GroupingFactoring a Four-Term Polynomial by Grouping1) Arrange the terms so that the first two terms have acommon factor and the last two terms have a commonfactor.2) For each pair of terms, use the distributive property tofactor out the pair’s greatest common factor.3) If there is now a common binomial factor, factor it out.4) If there is no common binomial factor in step 3, beginagain, rearranging the terms differently. If no rearrangement leads to a common binomialfactor, the polynomial cannot be factored.Martin-Gay, Developmental Mathematics24

Factoring by GroupingExampleFactor each of the following polynomials by grouping.1) x3 4x x2 4 x · x2 x · 4 1 · x2 1 · 4 x(x2 4) 1(x2 4) (x2 4)(x 1)2) 2x3 – x2 – 10x 5 x2 · 2x – x2 · 1 – 5 · 2x – 5 · (– 1) x2(2x – 1) – 5(2x – 1) (2x – 1)(x2 – 5)Martin-Gay, Developmental Mathematics25

Factoring by GroupingExampleFactor 2x – 9y 18 – xy by grouping.Neither pair has a common factor (other than 1).So, rearrange the order of the factors.2x 18 – 9y – xy 2 · x 2 · 9 – 9 · y – x · y 2(x 9) – y(9 x) 2(x 9) – y(x 9) (make sure the factors are identical)(x 9)(2 – y)Martin-Gay, Developmental Mathematics26

AssignmentWork on Unit 6A ReviewMartin-Gay, Developmental Mathematics27

Algebra 2 Exit QuizTuesday February 11. 2013Factor x2 11x – 42 0 and findthe zeros on the paper.Martin-Gay, Developmental Mathematics28

Dec 13, 2002 · Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Factor xy y 2x 2 by grouping. Notice that, although 1 is the GCF for all four terms of the polynomial, the first 2 terms have a GCF of y and the last 2 terms have a GCF of 2.