Factoring Polynomials

Factoring Polynomials

Chapter Sections 13.1 – The Greatest Common Factor 13.2 – Factoring Trinomials of the Form x2 bx c 13.3 – Factoring Trinomials of the Form ax2 bx c 13.4 – Factoring Trinomials of the Form x2 bx c by Grouping 13.5 – Factoring Perfect Square Trinomials and Difference of Two Squares 13.6 – Solving Quadratic Equations by Factoring 13.7 – Quadratic Equations and Problem Solving Martin-Gay, Developmental Mathematics 2

§ 13.1 The Greatest Common Factor

Factors Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. Factoring – writing a polynomial as a product of polynomials. Martin-Gay, Developmental Mathematics 4

Greatest Common Factor Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. Finding the GCF of a List of Integers or Terms 1) 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 Mathematics 5

Greatest Common Factor Example Find the GCF of each list of numbers. 1) 12 and 8 12 2 · 2 · 3 8 2·2·2 So the GCF is 2 · 2 4. 2) 7 and 20 7 1·7 20 2 · 2 · 5 There are no common prime factors so the GCF is 1. Martin-Gay, Developmental Mathematics 6

Greatest Common Factor Example Find the GCF of each list of numbers. 1) 6, 8 and 46 6 2·3 8 2·2·2 46 2 · 23 So the GCF is 2. 2) 144, 256 and 300 144 2 · 2 · 2 · 3 · 3 256 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 300 2 · 2 · 3 · 5 · 5 So the GCF is 2 · 2 4. Martin-Gay, Developmental Mathematics 7

Greatest Common Factor Example Find the GCF of each list of terms. 1) x3 and x7 x3 x · x · x x7 x · x · x · x · x · x · x So the GCF is x · x · x x3 t 6x5 and 4x3 6x5 2 · 3 · x · x · x 4x3 2 · 2 · x · x · x So the GCF is 2 · x · x · x 2x3 Martin-Gay, Developmental Mathematics 8

Greatest Common Factor Example Find the GCF of the following list of terms. a3b2, a2b5 and a4b7 a3b2 a · a · a · b · b a2b5 a · a · b · b · b · b · b a4b7 a · a · a · a · b · b · b · b · b · b · b So the GCF is a · a · b · b a2b2 Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable. Martin-Gay, Developmental Mathematics 9

Factoring Polynomials The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomial as a product by factoring out the GCF from all the terms. The remaining factors in each term will form a polynomial. Martin-Gay, Developmental Mathematics 10

Factoring out the GCF Example Factor out the GCF in each of the following polynomials. 1) 6x3 – 9x2 12x 3 · x · 2 · x2 – 3 · x · 3 · x 3 · x · 4 3x(2x2 – 3x 4) 2) 14x3y 7x2y – 7xy 7 · x · y · 2 · x2 7 · x · y · x – 7 · x · y · 1 7xy(2x2 x – 1) Martin-Gay, Developmental Mathematics 11

Factoring out the GCF Example Factor out the GCF in each of the following polynomials. 1) 6(x 2) – y(x 2) 6 · (x 2) – y · (x 2) (x 2)(6 – y) 2) xy(y 1) – (y 1) xy · (y 1) – 1 · (y 1) (y 1)(xy – 1) Martin-Gay, Developmental Mathematics 12

Factoring Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial. This will usually be followed by additional steps in the process. Example Factor 90 15y2 – 18x – 3xy2. 90 15y2 – 18x – 3xy2 3(30 5y2 – 6x – xy2) 3(5 · 6 5 · y2 – 6 · x – x · y2) 3(5(6 y2) – x (6 y2)) 3(6 y2)(5 – x) Martin-Gay, Developmental Mathematics 13

§ 13.2 Factoring Trinomials of the 2 Form x bx c

Factoring Trinomials Recall by using the FOIL method that F O I L (x 2)(x 4) x2 4x 2x 8 x2 6x 8 To factor x2 bx c into (x one #)(x another #), note that b is the sum of the two numbers and c is the product of the two numbers. So we’ll be looking for 2 numbers whose product is c and whose sum is b. Note: there are fewer choices for the product, so that’s why we start there first. Martin-Gay, Developmental Mathematics 15

Factoring Polynomials Example Factor the polynomial x2 13x 30. Since our two numbers must have a product of 30 and a sum of 13, the two numbers must both be positive. Positive factors of 30 Sum of Factors 1, 30 31 2, 15 17 3, 10 13 Note, there are other factors, but once we find a pair that works, we do not have to continue searching. So x2 13x 30 (x 3)(x 10). Martin-Gay, Developmental Mathematics 16

Factoring Polynomials Example Factor the polynomial x2 – 11x 24. Since our two numbers must have a product of 24 and a sum of -11, the two numbers must both be negative. Negative factors of 24 Sum of Factors – 1, – 24 – 25 – 2, – 12 – 14 – 3, – 8 – 11 So x2 – 11x 24 (x – 3)(x – 8). Martin-Gay, Developmental Mathematics 17

Factoring Polynomials Example Factor the polynomial x2 – 2x – 35. Since our two numbers must have a product of – 35 and a sum of – 2, the two numbers will have to have different signs. Factors of – 35 Sum of Factors – 1, 35 34 1, – 35 – 34 – 5, 7 2 5, – 7 –2 So x2 – 2x – 35 (x 5)(x – 7). Martin-Gay, Developmental Mathematics 18

Prime Polynomials Example Factor the polynomial x2 – 6x 10. Since our two numbers must have a product of 10 and a sum of – 6, the two numbers will have to both be negative. Negative factors of 10 Sum of Factors – 1, – 10 – 11 – 2, – 5 –7 Since there is not a factor pair whose sum is – 6, x2 – 6x 10 is not factorable and we call it a prime polynomial. Martin-Gay, Developmental Mathematics 19

Check Your Result! You should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial. Many times you can detect computational errors or errors in the signs of your numbers by checking your results. Martin-Gay, Developmental Mathematics 20

§ 13.3 Factoring Trinomials of 2 the Form ax bx c

Factoring Trinomials Returning to the FOIL method, F O I L (3x 2)(x 4) 3x2 12x 2x 8 3x2 14x 8 To factor ax2 bx c into (#1·x #2)(#3·x #4), note that a is the product of the two first coefficients, c is the product of the two last coefficients and b is the sum of the products of the outside coefficients and inside coefficients. Note that b is the sum of 2 products, not just 2 numbers, as in the last section. Martin-Gay, Developmental Mathematics 22

Factoring Polynomials Example Factor the polynomial 25x2 20x 4. 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 find a combination that works, or exhaust all of our possible pairs of factors. Keep in mind that, because some of our pairs are not identical factors, we may have to exchange some pairs of factors and make 2 attempts before we can definitely decide a particular pair of factors will not work. Continued. Martin-Gay, Developmental Mathematics 23

Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 20x. Factors Factors Resulting Product of Product of Sum of of 25x2 of 4 Binomials Outside Terms Inside Terms Products {x, 25x} {1, 4} (x 1)(25x 4) 4x 25x 29x (x 4)(25x 1) x 100x 101x {x, 25x} {2, 2} (x 2)(25x 2) 2x 50x 52x {5x, 5x} {2, 2} (5x 2)(5x 2) 10x 10x 20x Continued. Martin-Gay, Developmental Mathematics 24

Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (5x 2)(5x 2) 5x(5x) 5x(2) 2(5x) 2(2) 25x2 10x 10x 4 25x2 20x 4 So our final answer when asked to factor 25x2 20x 4 will be (5x 2)(5x 2) or (5x 2)2. Martin-Gay, Developmental Mathematics 25

Factoring Polynomials Example Factor the polynomial 21x2 – 41x 10. Possible factors of 21x2 are {x, 21x} or {3x, 7x}. Since the middle term is negative, possible factors of 10 must both be negative: {-1, -10} or {-2, -5}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Continued. Martin-Gay, Developmental Mathematics 26

Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 41x. Factors Factors Resulting Product of Product of Sum of of 21x2 of 10 Binomials Outside Terms Inside Terms Products {x, 21x}{1, 10}(x – 1)(21x – 10) –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 Martin-Gay, Developmental Mathematics – 107x Continued. 27

Factoring Polynomials Example Continued Factors Factors Resulting Product of Product of Sum of of 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 41x Continued. Martin-Gay, Developmental Mathematics 28

Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (3x – 5)(7x – 2) 3x(7x) 3x(-2) - 5(7x) - 5(-2) 21x2 – 6x – 35x 10 21x2 – 41x 10 So our final answer when asked to factor 21x2 – 41x 10 will be (3x – 5)(7x – 2). Martin-Gay, Developmental Mathematics 29

Factoring Polynomials Example Factor the polynomial 3x2 – 7x 6. The only possible factors for 3 are 1 and 3, so we know that, if factorable, the polynomial will have to look like (3x )(x ) in factored form, so that the product of the first two terms in the binomials will be 3x2. Since the middle term is negative, possible factors of 6 must both be negative: { 1, 6} or { 2, 3}. We need to methodically try each pair of factors until we find a combination that works, or exhaust all of our possible pairs of factors. Continued. Martin-Gay, Developmental Mathematics 30

Factoring Polynomials Example Continued We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to 7x. Factors of 6 Resulting Binomials { 1, 6} (3x – 1)(x – 6) { 2, 3} Product of Product of Sum of Outside Terms Inside Terms Products 18x x 19x (3x – 6)(x – 1) Common factor so no need to test. (3x – 2)(x – 3) 9x (3x – 3)(x – 2) Common factor so no need to test. 2x 11x Continued. Martin-Gay, Developmental Mathematics 31

Factoring Polynomials Example Continued Now we have a problem, because we have exhausted all possible choices for the factors, but have not found a pair where the sum of the products of the outside terms and the inside terms is –7. So 3x2 – 7x 6 is a prime polynomial and will not factor. Martin-Gay, Developmental Mathematics 32

Factoring Polynomials Example Factor the polynomial 6x2y2 – 2xy2 – 60y2. Remember that the larger the coefficient, the greater the probability of having multiple pairs of factors to check. So it is important that you attempt to factor out any common factors first. 6x2y2 – 2xy2 – 60y2 2y2(3x2 – x – 30) The only possible factors for 3 are 1 and 3, so we know that, if we can factor the polynomial further, it will have to look like 2y2(3x )(x ) in factored form. Continued. Martin-Gay, Developmental Mathematics 33

Factoring Polynomials Example Continued Since the product of the last two terms of the binomials will have to be –30, we know that they must be different signs. Possible factors of –30 are {–1, 30}, {1, –30}, {–2, 15}, {2, –15}, {–3, 10}, {3, –10}, {–5, 6} or {5, –6}. We will be looking for a combination that gives the sum of the products of the outside terms and the inside terms equal to –x. Continued. Martin-Gay, Developmental Mathematics 34

Factoring Polynomials Example Continued Factors Resulting Product of Product of Sum of of -30 Binomials Outside Terms Inside Terms Products {-1, 30} (3x – 1)(x 30) 90x -x 89x (3x 30)(x – 1) {1, -30} (3x 1)(x – 30) (3x – 30)(x 1) {-2, 15} (3x – 2)(x 15) (3x 15)(x – 2) {2, -15} (3x 2)(x – 15) (3x – 15)(x 2) Common factor so no need to test. -90x x -89x Common factor so no need to test. 45x -2x 43x Common factor so no need to test. -45x 2x -43x Common factor so no need to test. Continued. Martin-Gay, Developmental Mathematics 35

Factoring Polynomials Example Continued Factors of –30 {–3, 10} Resulting Binomials (3x – 3)(x 10) (3x 10)(x – 3) {3, –10} (3x 3)(x – 10) (3x – 10)(x 3) Product of Product of Sum of Outside Terms Inside Terms Products Common factor so no need to test. –9x 10x x Common factor so no need to test. 9x –10x –x Continued. Martin-Gay, Developmental Mathematics 36

Factoring Polynomials Example Continued Check the resulting factorization using the FOIL method. F O I L (3x – 10)(x 3) 3x(x) 3x(3) – 10(x) – 10(3) 3x2 9x – 10x – 30 3x2 – x – 30 So our final answer when asked to factor the polynomial 6x2y2 – 2xy2 – 60y2 will be 2y2(3x – 10)(x 3). Martin-Gay, Developmental Mathematics 37

§ 13.4 Factoring Trinomials of 2 the Form x bx c by Grouping

Factoring by Grouping Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Example 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. 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 Mathematics 39

Factoring by Grouping Factoring a Four-Term Polynomial by Grouping 1) 2) 3) 4) Arrange the terms so that the first two terms have a common factor and the last two terms have a common factor. For each pair of terms, use the distributive property to factor out the pair’s greatest common factor. If there is now a common binomial factor, factor it out. If there is no common binomial factor in step 3, begin again, rearranging the terms differently. If no rearrangement leads to a common binomial factor, the polynomial cannot be factored. Martin-Gay, Developmental Mathematics 40

Factoring by Grouping Example Factor 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 Mathematics 41

Factoring by Grouping Example Factor 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 Mathematics 42

§ 13.5 Factoring Perfect Square Trinomials and the Difference of Two Squares

Perfect Square Trinomials Recall that in our very first example in Section 4.3 we attempted to factor the polynomial 25x2 20x 4. The result was (5x 2)2, an example of a binomial squared. Any trinomial that factors into a single binomial squared is called a perfect square trinomial. Martin-Gay, Developmental Mathematics 44

Perfect Square Trinomials In the last chapter we learned a shortcut for squaring a binomial (a b)2 a2 2ab b2 (a – b)2 a2 – 2ab b2 So if the first and last terms of our polynomial to be factored are can be written as expressions squared, and the middle term of our polynomial is twice the product of those two expressions, then we can use these two previous equations to easily factor the polynomial. a2 2ab b2 (a b)2 a2 – 2ab b2 (a – b)2 Martin-Gay, Developmental Mathematics 45

Perfect Square Trinomials Example Factor the polynomial 16x2 – 8xy y2. Since the first term, 16x2, can be written as (4x)2, and the last term, y2 is obviously a square, we check the middle term. 8xy 2(4x)(y) (twice the product of the expressions that are squared to get the first and last terms of the polynomial) Therefore 16x2 – 8xy y2 (4x – y)2. Note: You can use FOIL method to verify that the factorization for the polynomial is accurate. Martin-Gay, Developmental Mathematics 46

Difference of Two Squares Another shortcut for factoring a trinomial is when we want to factor the difference of two squares. a2 – b2 (a b)(a – b) A binomial is the difference of two square if 1.both terms are squares and 2.the signs of the terms are different. 9x2 – 25y2 – c4 d4 Martin-Gay, Developmental Mathematics 47

Difference of Two Squares Example Factor the polynomial x2 – 9. The first term is a square and the last term, 9, can be written as 32. The signs of each term are different, so we have the difference of two squares Therefore x2 – 9 (x – 3)(x 3). Note: You can use FOIL method to verify that the factorization for the polynomial is accurate. Martin-Gay, Developmental Mathematics 48

§ 13.6 Solving Quadratic Equations by Factoring

Zero Factor Theorem Quadratic Equations Can be written in the form ax2 bx c 0. a, b and c are real numbers and a 0. This is referred to as standard form. Zero Factor Theorem If a and b are real numbers and ab 0, then a 0 or b 0. This theorem is very useful in solving quadratic equations. Martin-Gay, Developmental Mathematics 50

Solving Quadratic Equations Steps for Solving a Quadratic Equation by Factoring 1) 2) 3) 4) 5) Write the equation in standard form. Factor the quadratic completely. Set each factor containing a variable equal to 0. Solve the resulting equations. Check each solution in the original equation. Martin-Gay, Developmental Mathematics 51

Solving Quadratic Equations Example Solve x2 – 5x 24. First write the quadratic equation in standard form. x2 – 5x – 24 0 Now we factor the quadratic using techniques from the previous sections. x2 – 5x – 24 (x – 8)(x 3) 0 We set each factor equal to 0. x – 8 0 or x 3 0, which will simplify to x 8 or x – 3 Continued. Martin-Gay, Developmental Mathematics 52

Solving Quadratic Equations Example Continued Check both possible answers in the original equation. 82 – 5(8) 64 – 40 24 true (–3)2 – 5(–3) 9 – (–15) 24 true So our solutions for x are 8 or –3. Martin-Gay, Developmental Mathematics 53

Solving Quadratic Equations Example Solve 4x(8x 9) 5 First write the quadratic equation in standard form. 32x2 36x 5 32x2 36x – 5 0 Now we factor the quadratic using techniques from the previous sections. 32x2 36x – 5 (8x – 1)(4x 5) 0 We set each factor equal to 0. 8x – 1 0 or 4x 5 0 1 5 8x 1 or 4x – 5, which simplifies to x8 or . 4 Continued. Martin-Gay, Developmental Mathematics 54

Solving Quadratic Equations Example Continued Check both possible answers in the original equation. 1 4 8 1 1 8 9 4 8 8 1 1 1 9 4 8 (10) 2 (10) 5 5 5 5 5 8 9 4 10 9 4 ( 1) ( 5)( 1) 5 4 4 4 4 4 true true So our solutions for x are 1 8 or 5 . 4 Martin-Gay, Developmental Mathematics 55

Finding x-intercepts Recall that in Chapter 3, we found the x-intercept of linear equations by letting y 0 and solving for x. The same method works for x-intercepts in quadratic equations. Note: When the quadratic equation is written in standard form, the graph is a parabola opening up (when a 0) or down (when a 0), where a is the coefficient of the x2 term. The intercepts will be where the parabola crosses the xaxis. Martin-Gay, Developmental Mathematics 56

Finding x-intercepts Example Find the x-intercepts of the graph of y 4x2 11x 6. The equation is already written in standard form, so we let y 0, then factor the quadratic in x. 0 4x2 11x 6 (4x 3)(x 2) We set each factor equal to 0 and solve for x. 4x 3 0 or x 2 0 4x –3 or x –2 x –¾ or x –2 So the x-intercepts are the points (–¾, 0) and (–2, 0). Martin-Gay, Developmental Mathematics 57

§ 13.7 Quadratic Equations and Problem Solving

Strategy for Problem Solving General Strategy for Problem Solving 1) 2) 3) 4) Understand the problem Read and reread the problem Choose a variable to represent the unknown Construct a drawing, whenever possible Propose a solution and check Translate the problem into an equation Solve the equation Interpret the result Check proposed solution in problem State your conclusion Martin-Gay, Developmental Mathematics 59

Finding an Unknown Number Example The product of two consecutive positive integers is 132. Find the two integers. 1.) Understand Read and reread the problem. If we let x one of the unknown positive integers, then x 1 the next consecutive positive integer. Continued Martin-Gay, Developmental Mathematics 60

Finding an Unknown Number Example continued 2.) Translate The product of two consecutive positive integers x (x 1) is 132 132 Continued Martin-Gay, Developmental Mathematics 61

Finding an Unknown Number Example continued 3.) Solve x(x 1) 132 x2 x 132 x2 x – 132 0 (Distributive property) (Write quadratic in standard form) (x 12)(x – 11) 0 (Factor quadratic polynomial) x 12 0 or x – 11 0 x –12 or x 11 (Set factors equal to 0) (Solve each factor for x) Continued Martin-Gay, Developmental Mathematics 62

Finding an Unknown Number Example continued 4.) Interpret Check: Remember that x is suppose to represent a positive integer. So, although x -12 satisfies our equation, it cannot be a solution for the problem we were presented. If we let x 11, then x 1 12. The product of the two numbers is 11 · 12 132, our desired result. State: The two positive integers are 11 and 12. Martin-Gay, Developmental Mathematics 63

The Pythagorean Theorem Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. (leg a)2 (leg b)2 (hypotenuse)2 leg a hypotenuse leg b Martin-Gay, Developmental Mathematics 64

The Pythagorean Theorem Example Find the length of the shorter leg of a right triangle if the longer leg is 10 miles more than the shorter leg and the hypotenuse is 10 miles less than twice the shorter leg. 1.) Understand Read and reread the problem. If we let x the length of the shorter leg, then x x 10 the length of the longer leg and 2 x - 10 x 10 2x – 10 the length of the hypotenuse. Continued Martin-Gay, Developmental Mathematics 65

The Pythagorean Theorem Example continued 2.) Translate By the Pythagorean Theorem, (leg a)2 (leg b)2 (hypotenuse)2 x2 (x 10)2 (2x – 10)2 3.) Solve x2 (x 10)2 (2x – 10)2 x2 x2 20x 100 4x2 – 40x 100 2x2 20x 100 4x2 – 40x 100 0 2x2 – 60x 0 2x(x – 30) x 0 or x 30 (multiply the binomials) (simplify left side) (subtract 2x2 20x 100 from both sides) (factor right side) (set each factor 0 and solve) Continued Martin-Gay, Developmental Mathematics 66

The Pythagorean Theorem Example continued 4.) Interpret Check: Remember that x is suppose to represent the length of the shorter side. So, although x 0 satisfies our equation, it cannot be a solution for the problem we were presented. If we let x 30, then x 10 40 and 2x – 10 50. Since 302 402 900 1600 2500 502, the Pythagorean Theorem checks out. State: The length of the shorter leg is 30 miles. (Remember that is all we were asked for in this problem.) Martin-Gay, Developmental Mathematics 67