Factoring by Grouping Section 8-8

Goals Goal To factor higher degree polynomials by grouping. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Factoring by Grouping

Definition Factoring by Grouping –Using the distributive property to factor polynomials with four or more terms. –terms can be put into groups and then factored---- each group will have a “like” factor used in regrouping.

Factoring by Grouping Polynomials with four or more terms like 3xy – 21y + 5x – 35, can sometimes be factored by grouping terms of the polynomials. One method is to group the terms into binomials that can be factored using the distributive property. Then use the distributive property again with a binomial as the common factor.

A polynomial can be factored by grouping if all of the following conditions exist. 1.There are four or more terms. 2.Terms have common factors that can be grouped together, and 3.There are two common factors that are identical. Symbols:ax + bx + ay + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (x + y)(a + b) Group, factor Regroup Factor by Grouping

Factor each polynomial by grouping. Check your answer. 6h 4 – 4h h – 8 (6h 4 – 4h 3 ) + (12h – 8) 2h 3 (3h – 2) + 4(3h – 2) (3h – 2)(2h 3 + 4) Group terms that have a common number or variable as a factor. Factor out the GCF of each group. (3h – 2) is another common factor. Factor out (3h – 2). Example: Factor by Grouping

Check your answer. Check (3h – 2)(2h 3 + 4) Multiply to check your solution. 3h(2h 3 ) + 3h(4) – 2(2h 3 ) – 2(4) 6h h – 4h 3 – 8 The product is the original polynomial. 6h 4 – 4h h – 8 Example: Continued

Factor each polynomial by grouping. 5y 4 – 15y 3 + y 2 – 3y (5y 4 – 15y 3 ) + (y 2 – 3y) 5y 3 (y – 3) + y(y – 3) (y – 3)(5y 3 + y) Group terms. Factor out the GCF of each group. (y – 3) is a common factor. Factor out (y – 3). Example: Factor by Grouping

Factor each polynomial by grouping. 6b 3 + 8b 2 + 9b + 12 (6b 3 + 8b 2 ) + (9b + 12) 2b 2 (3b + 4) + 3(3b + 4) (3b + 4)(2b 2 + 3) Group terms. Factor out the GCF of each group. (3b + 4) is a common factor. Factor out (3b + 4). Your Turn:

Factor each polynomial by grouping. 4r r + r (4r r) + (r 2 + 6) 4r(r 2 + 6) + 1(r 2 + 6) (r 2 + 6)(4r + 1) Group terms. Factor out the GCF of each group. (r 2 + 6) is a common factor. Factor out (r 2 + 6). Your Turn:

If two quantities are opposites, their sum is 0 (Additive Inverses). (5 – x) + (x – 5) 5 – x + x – 5 – x + x + 5 – Helpful Hint

Recognizing opposite binomials can help you factor polynomials. The binomials (5 – x) and (x – 5) are opposites. Notice (5 – x) can be written as – 1(x – 5). – 1(x – 5) = ( – 1)(x) + ( – 1)( – 5) = – x + 5 = 5 – x So, (5 – x) = – 1(x – 5) Distributive Property. Simplify. Commutative Property of Addition. Additive Inverse

Factor 2x 3 – 12x – 3x 2x 3 – 12x – 3x (2x 3 – 12x 2 ) + (18 – 3x) 2x 2 (x – 6) + 3(6 – x) 2x 2 (x – 6) + 3( – 1)(x – 6) 2x 2 (x – 6) – 3(x – 6) (x – 6)(2x 2 – 3) Group terms. Factor out the GCF of each group. Simplify. (x – 6) is a common factor. Factor out (x – 6). Write (6 – x) as –1(x – 6). Example: Factoring with Opposite Groups

Factor each polynomial. Check your answer. 15x 2 – 10x 3 + 8x – 12 (15x 2 – 10x 3 ) + (8x – 12) 5x 2 (3 – 2x) + 4(2x – 3) 5x 2 (3 – 2x) + 4( – 1)(3 – 2x) 5x 2 (3 – 2x) – 4(3 – 2x) (3 – 2x)(5x 2 – 4) Group terms. Factor out the GCF of each group. Simplify. (3 – 2x) is a common factor. Factor out (3 – 2x). Write (2x – 3) as –1(3 – 2x). Your Turn:

Factor each polynomial. Check your answer. 8y – 8 – x + xy (8y – 8) + ( – x + xy) 8(y – 1)+ (x)( – 1 + y) (y – 1)(8 + x) 8(y – 1)+ (x)(y – 1) Group terms. Factor out the GCF of each group. (y – 1) is a common factor. Factor out (y – 1). Your Turn:

Factor each polynomial by grouping. 1. 2x 3 + x 2 – 6x – p 4 – 2p p – 18 (7p – 2)(p 3 + 9) (2x + 1)(x 2 – 3) Your Turn:

Factoring Procedure

Methods of Factoring

Assignment 8-8 Exercises Pg : #10 – 50 even