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Factoring Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Greatest Common Factor The simplest method.

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Presentation on theme: "Factoring Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Greatest Common Factor The simplest method."— Presentation transcript:

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 x. GCF = 6x 18x x = 6x (3x 2 ) + 6x (10) 18x 3 = 2 · 3 · 3 · x · x · x Apply the distributive law to factor the polynomial. 6x (3x ) = 6x (3x 2 ) + 6x (10) = 18x x Check the answer by multiplication. Factor each term. Find the GCF. 60x = 2 · 2 · 3 · 5 · x = 6x (3x ) = (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 GCF = 4. 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) = (x + 1) (5 – y) (x + 1) (5 – y) = 5(x + 1) – y(x + 1) Check. Example: Factor = 4(x 2 – 3x + 5)

4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 A difference of two 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 (x + 1) 2 – 25y 4. = (x + 1) 2 – (5y 2 ) 2 = [(x + 1) + (5y 2 )][(x + 1) – (5y 2 )] = (x y 2 )(x + 1 – 5y 2 ) Difference of Squares a 2 – b 2 = (a + b)(a – b). x 2 – 9y 2 (x + 1) 2 – 25y 4 D.O.T.S.

5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved 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 = y (2x + 3) – 2(2x + 3) = (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). = (a – b) (2a – 3c) Factor. Examples: Factor 2xy + 3y – 4x – 6 2a 2 + 3bc – 2ab – 3ac Notice the sign!

6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Factor x 2 + bx + c Factoring these trinomials is based on reversing the FOIL process. To factor a simple trinomial of the form x 2 + bx + c, express the trinomial as the product of two binomials. For example, x x + 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 + (b + a) x + ba 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 + bx+ ax+ ba x 2 + 3x + 2 (Product-sum method)

7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Factor Example: Factor x 2 – 8x = (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 SumNegative Factors of 15 -3, , and ab = 15. x 2 – 8x + 15

8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Factor Example: Factor x x = (x + a)(x + b) Check: (x + 4)(x + 9) Therefore a and b are: x x + 36 = x 2 + 9x + 4x + 36= x x = (x + 4)(x + 9) = x 2 + (a + b) x + ab SumPositive Factors of 36 1, , 12 4, , , 1820 x x + 36 two positive factors of 36 whose sum is

9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Factor Completely Example: Factor 4x 3 – 40x x. A polynomial is factored completely when it is written as a product of factors that can not be factored further. 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 x 4x 3 – 40x x Check:

10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Factoring Polynomials of the Form ax 2 + bx + c Factoring complex trinomials of the form ax 2 + bx + c, (a 1) can be done by decomposition or cross-check method. Example: Factor 3x 2 + 8x Find the product of first and last terms 3 4 = 12 Decomposition Method 2. We need to find factors of 12 whose sum is 8 1, 12 2, 6 3, 4 3. Rewrite the middle term decomposed into the two numbers 3x 2 + 2x + 6x + 4 = x(3x + 2) + 2(3x + 2) = (3x 2 + 2x) + (6x + 4) 4. Factor by grouping in pairs = (3x + 2) (x + 2) 3x 2 + 8x + 4 = (3x + 2) (x + 2)

11 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Factor Example: Factor 4x 2 + 8x – 5. 4x 2 + 8x – 5 = (2x –1)(2x – 5) 4 5 = 20 We need to find factors of 20 whose difference is 8 1, 20 2, 10 4, 5 Rewrite the middle term decomposed into the two numbers 4x 2 – 2x + 10x – 5 = 2x(2x – 1) + 5(2x – 1) = (4x 2 – 2x) + (10x – 5) = (2x – 1) (2x + 5) Factor by grouping in pairs


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