1. 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

2. 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

3. 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

4. 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

5. 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

6. 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