Factor out a common binomial

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 6.1 The Greatest Common Factor and Factoring by Grouping Copyright © 2013, 2009, 2006 Pearson.

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Presentation transcript:

Factor out a common binomial EXAMPLE 1 Factor out a common binomial Factor the expression. 2x(x + 4) – 3(x + 4) a. 3y2(y – 2) + 5(2 – y) b. SOLUTION 2x(x + 4) – 3(x + 4) = (x + 4)(2x – 3) a. The binomials y – 2 and 2 – y are opposites. Factor – 1 from 2 – y to obtain a common binomial factor. b. 3y2(y – 2) + 5(2 – y) = 3y2(y – 2) – 5(y – 2) Factor – 1 from (2 – y). = (y – 2)(3y2 – 5) Distributive property

y2 + y + yx + x = (y2 + y) + (yx + x) b. = y(y + 1) + x(y + 1) EXAMPLE 2 Factor by grouping Factor the polynomial. x3 + 3x2 + 5x + 15. a y2 + y + yx + x b. SOLUTION x3 + 3x2 + 5x + 15 = (x3 + 3x2) + (5x + 15) a. Group terms. = x2(x + 3) + 5(x + 3) Factor each group. = (x + 3)(x2 + 5) Distributive property y2 + y + yx + x = (y2 + y) + (yx + x) b. Group terms. = y(y + 1) + x(y + 1) Factor each group. = (y + 1)(y + x) Distributive property

EXAMPLE 3 Factor by grouping Factor 6 + 2x . x3 – 3x2 SOLUTION The terms x and – 6 have no common factor. Use the commutative property to rearrange the terms so that you can group terms with a common factor. 3 x3– 6 +2x – 3x2 = x3– 3x2 +2x – 6 Rearrange terms. (x3 – 3x2 ) + (2x – 6) = Group terms. x2 (x – 3 ) + 2(x – 3) = Factor each group. (x – 3 ) (x2+ 2) = Distributive property

Check your factorization using a graphing calculator. Graph y and y EXAMPLE 3 Factor by grouping CHECK Check your factorization using a graphing calculator. Graph y and y Because the graphs coincide, you know that your factorization is correct. 1 = (x – 3)(x2 + 2) . 2 6 + 2x = x3 – – 3x2

GUIDED PRACTICE for Examples 1, 2 and 3 Factor the expression. 1. x (x – 2) + (x – 2) x (x – 2) + (x – 2) = x (x – 2) + 1(x – 2) Factor 1 from x – 2. = (x – 2) (x + 1) Distributive property 2. a3 + 3a2 + a + 3. a3 + 3a2 + a + 3 = (a3 + 3a2) + (a + 3) Group terms. = a2(a + 3) + 1(a + 3) Factor each group. = (a2 + 1)(a + 3) Distributive property

GUIDED PRACTICE for Examples 1, 2 and 3 3. y2 + 2x + yx + 2y. SOLUTION The terms y2 and 2x have no common factor. Use the commutative property to rearrange the terms so that you can group terms with a common factor. y2 + 2x + yx + 2y = y2 + yx + 2y +2x Rearrange terms. = ( y2 + yx ) +( 2y +2x ) Group terms. = y( y + x ) + 2(y +x ) Factor each group. = (y + 2)( y + x ) Distributive property