Factoring by GCFFactoring by GCF

FactoringFactoring

Put the expression in a division towerPut the expression in a division towerContinue to divide by numbers or Continue to divide by numbers or variables until there is no number or variables until there is no number or variable common to all terms.variable common to all terms.Put the numbers and variables along Put the numbers and variables along the side on the outside of the the side on the outside of the parentheses.parentheses.Put the top expression on the inside of Put the top expression on the inside of parentheses.parentheses.

Example 1Example 156x4 – 32x3 – 72x2

Factor each expression.

A. 5(x + 2) + 3x(x + 2)

5(x + 2) + 3x(x + 2)

(x + 2)(3x + 5)

The terms have a common binomial factor of (x + 2).

Factor out (x + 2).

B. –2b(b2 + 1)+ (b2 + 1)

–2b(b2 + 1) + (b2 + 1)

–2b(b2 + 1) + 1(b2 + 1)

(b2 + 1)(–2b + 1)

The terms have a common binomial factor of (b2 + 1).

(b2 + 1) = 1(b2 + 1)

Factor out (b2 + 1).

Example 2: Factoring Out a Example 2: Factoring Out a Common Binomial FactorCommon Binomial Factor

Example 3: Factoring by GroupingExample 3: Factoring by Grouping

Factor each polynomial by grouping. Check your answer.

6h4 – 4h3 + 12h – 8

(6h4 – 4h3) + (12h – 8)

2h3(3h – 2) + 4(3h – 2)

2h3(3h – 2) + 4(3h – 2)

2(3h – 2)(h3 + 2)

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 4: Factoring by GroupingExample 4: Factoring by Grouping

Factor each polynomial by grouping. Check your answer.

5y4 – 15y3 + y2 – 3y

(5y4 – 15y3) + (y2 – 3y)

5y3(y – 3) + y(y – 3)

5y3(y – 3) + y(y – 3)

y(y – 3)(5y2 + 1)

Group terms.

Factor out the GCF of each group.

(y – 3) is a common factor.

Factor out (y – 3).

Example 5: Factoring with Example 5: Factoring with OppositesOpposites

Factor 2x3 – 12x2 + 18 – 3x

2x3 – 12x2 + 18 – 3x

(2x3 – 12x2) + (18 – 3x)

2x2(x – 6) + 3(6 – x)

2x2(x – 6) + 3(–1)(x – 6)

2x2(x – 6) – 3(x – 6)

(x – 6)(2x2 – 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 6Example 6

Factor each polynomial. Check your answer.

15x2 – 10x3 + 8x – 12

(15x2 – 10x3) + (8x – 12)

5x2(3 – 2x) + 4(2x – 3)

5x2(3 – 2x) + 4(–1)(3 – 2x)

5x2(3 – 2x) – 4(3 – 2x)

-1(2x - 3)(5x2 – 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).

Try these…Try these…

Factor each polynomial. Check your answer.

1. 16x + 20x3

2. 4m4 – 12m2 + 8m

Factor each expression.

3. 7k(k – 3) + 4(k – 3)

4. 3y(2y + 3) – 5(2y + 3)

(2y + 3)(3y – 5)

(k – 3)(7k + 4)

4m(m3 – 3m + 2)

4x(4 + 5x2)

Try these (cont)…Try these (cont)…Factor each polynomial by grouping. Check your answer.

5. 2x3 + x2 – 6x – 3

6. 7p4 – 2p3 + 63p – 18

7. A rocket is fired vertically into the air at 40 m/s.

The expression –5t2 + 40t + 20 gives the

rocket’s height after t seconds. Factor this

expression.

–5(t2 – 8t – 4)

(7p – 2)(p3 + 9)

(2x + 1)(x2 – 3)