1. Factoring Polynomials Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Greatest Common Factor The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term. Example: Factor 18x 3 + 60x. GCF = 6x 18x 3 + 60x = 6x (3x 2 ) + 6x (10) 18x 3 = 2 · 3 · 3 · x · x · x Apply the distributive law to factor the polynomial. 6x (3x 2 + 10) = 6x (3x 2 ) + 6x (10) = 18x 3 + 60x Check the answer by multiplication. Factor each term. Find the GCF. 60x = 2 · 2 · 3 · 5 · x = 6x (3x 2 + 10) = (2 · 3 · x) · 3 · x · x = (2 · 3 · x) · 2 · 5

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example: Factor 4x 2 – 12x + 20. 4x 2 = 2 · 2 · x · x Therefore, GCF = 4. 4x 2 – 12x + 20 = 4x 2 – 4 · 3x + 4 · 5 4(x 2 – 3x + 5) = 4x 2 – 12x + 20 Check the answer. A common binomial factor can be factored out of certain expressions. Example: Factor the expression 5(x + 1) – y(x + 1). 5(x + 1) – y(x + 1) = (5 – y)(x + 1) (5 – y)(x + 1) = 5(x + 1) – y(x + 1) Check. Example: Factor = 4(x 2 – 3x + 5), 12x = 2 · 2 · 3 · x, 20 = 2 · 2 · 5

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 A difference of squares can be factored using the formula Example: Factor x 2 – 9y 2. = (x) 2 – (3y) 2 = (x + 3y)(x – 3y) Write terms as perfect squares. Use the formula. The same method can be used to factor any expression which can be written as a difference of squares. Example: Factor 4(x + 1) 2 – 25y 4. = (2(x + 1)) 2 – (5y 2 ) 2 = [(2(x + 1)) + (5y 2 )][(2(x + 1)) – (5y 2 )] = (2x + 2 + 5y 2 )(2x + 2 – 5y 2 ) Difference of Squares a 2 – b 2 = (a + b)(a – b). x 2 – 9y 2 4(x + 1) 2 – 25y 4

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 2. Factor 2a 2 + 3bc – 2ab – 3ac. Some polynomials can be factored by grouping terms to produce a common binomial factor. = 2a 2 – 2ab + 3bc – 3ac = (2x + 3)y – (2x + 3)2 = (2a 2 – 2ab) + (3bc – 3ac) = 2a(a – b) + 3c(b – a) = (2xy + 3y) – (4x + 6) Group terms. Examples: 1. Factor 2xy + 3y – 4x – 6. Factor each pair of terms. = (2x + 3)( y – 2) Factor out the common binomial. Rearrange terms. Group terms. Factor. = 2a(a – b) – 3c(a – b) b – a = – (a – b). = (2a – 3c)(a – b) Factor. Examples: Factor 2xy + 3y – 4x – 6 2a 2 + 3bc – 2ab – 3ac

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 A sum or difference of cubes can be factored using the formulas Examples: 1. Factor x 3 + 8y 3. 2. Factor 27a 3 – y 3 z 6. = (x) 3 + (2y) 3 = (x + 2y)(x 2 – x(2y) + (2y) 2 ) Use the sum formula. Write each term as a perfect cube. Use the difference formula. = (x + 2y)(x 2 – 2xy + 4y 2 ) Simplify. = (3a) 3 – ( yz 2 ) 3 Simplify. = ((3a) – ( yz 2 ))((3a) 2 + (3a)( yz 2 ) +( yz 2 ) 2 ) = (3a – yz 2 )(9a 2 + 3ayz 2 + y 2 z 4 ) Sum or Difference of Cubes a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) and a 3 – b 3 = (a – b)(a 2 + ab + b 2 ). x 3 + 8y 3 27a 3 – y 3 z 6

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Factor x 2 + bx + c One method of factoring trinomials is based on reversing the FOIL process. To factor a trinomial of the form x 2 + bx + c, express the trinomial as the product of two binomials. For example, x 2 + 10x + 24 = (x + 4)(x + 6). Example: Factor x 2 + 3x + 2. = (x + a)(x + b) Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b. = x 2 F Apply FOIL to multiply the binomials. = x 2 + (a + b) x + ab Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2. = x 2 + (1 + 2) x + 1 · 2 Therefore, x 2 + 3x + 2 = (x + 1)(x + 2). OIL + ax+ bx+ ab x 2 + 3x + 2

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Factor Example: Factor x 2 – 8x + 15. = (x + a)(x + b) (x – 3)(x – 5) = x 2 – 5x – 3x + 15 x 2 – 8x + 15 = (x – 3)(x – 5). Therefore a + b = – 8 Check: = x 2 + (a + b)x + ab It follows that both a and b are negative. = x 2 – 8x + 15. SumNegative Factors of 15 – 3, – 5– 8– 8 – 1, – 15+15 and ab = 15. x 2 – 8x + 15

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Factor Example: Factor x 2 + 13x + 36. = (x + a)(x + b) Check: (x + 4)(x + 9) Therefore a and b are two positive factors of 36 whose sum is 13. x 2 + 13x + 36 = x 2 + 9x + 4x + 36= x 2 + 13x + 36. = (x + 4)(x + 9) = x 2 + (a + b) x + ab SumPositive Factors of 36 1, 3637 15153, 12 4, 91313 6, 61212 2, 1820 x 2 + 13x + 36

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Factor Completely Example: Factor 4x 3 – 40x 2 + 100x. A polynomial is factored completely when it is written as a product of factors that can not be factored further. = 4x(x 2 ) – 4x(10x) + 4x(25) The GCF is 4x. = 4x(x 2 – 10x + 25) Use distributive property to factor out the GCF. = 4x(x – 5)(x – 5) Factor the trinomial. 4x(x – 5)(x – 5)= 4x(x 2 – 5x – 5x + 25) = 4x(x 2 – 10x + 25) = 4x 3 – 40x 2 + 100x 4x 3 – 40x 2 + 100x Check:

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Factoring Polynomials of the Form ax 2 + bx + c Factoring polynomials of the form ax 2 + bx + c, (a  1) is usually done by trial and error. Example: Factor 2x 2 + 5x + 3. = (2x + a)(x + b) = 2x 2 + (a + 2b)x + ab For some a and b. Therefore, a + 2b = 5 and ab = 3. 2x 2 + 5x + 3 = (2x + 3)(x + 1) Check: (2x + 3)(x + 1) = 2x 2 + 2x + 3x + 3 = 2x 2 + 5x + 3. a b a + 2b 1 3 7 3 15 – 1 – 3 – 7 – 3 – 1 – 5 2x 2 + 5x + 3

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Factor Example: Factor 4x 2 – 12x + 5. This polynomial factors as (x + a)(4x + b) or (2x + a)(2x + b). Create trial factorizations with the forms above. 4x 2 – 12x + 5 = (2x –1)(2x – 5) The middle term –12x equals either (4a + b) x or (2a + 2b) x. Since a and b cannot both be positive, they must both be negative. Since ab = + 5, a and b have the same sign. a = –1, b = – 5 or a = 1 and b = 5. Trial FactorsMiddle Term (x –1)(4x – 5)– 5x – 4x = – 9x (2x –1)(2x – 5)– 10x – 2x = –12x (x –5)(4x – 1)– x – 20x = – 21x

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Factoring Trinomials Trinomials which are quadratic in form are factored like quadratic trinomials. Example: Factor 3x 4 + 28x 2 + 9. = 3u 2 + 28u + 9 = (3x 2 + 1)(x 2 + 9) = (3u + 1)(u + 9) Let u = x 2. Factor. Replace u by x 2. Many trinomials cannot be factored. Example: Factor x 2 + 3x + 5. Let x 2 + 3x + 5 = (x + a)(x + b) = x 2 + (a + b) x + ab. The trinomial x 2 + 3x + 5 cannot be factored. Then a + b = 3 and ab = 5. This is impossible. 3x 4 + 28x 2 + 9