Factoring by Grouping

Factoring by
Grouping
Objective
I will factor four term polynomials by
using the grouping method.
Example 3: Factoring Out a Common Binomial Factor
Factor each expression.
A. 5(x + 2) + 3x(x + 2)
5(x + 2) + 3x(x + 2)
(x + 2)(5 + 3x)
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) The terms have a common
binomial factor of (b2 + 1).
–2b(b2 + 1) + 1(b2 + 1) (b2 + 1) = 1(b2 + 1)
(b2 + 1)(–2b + 1)
Factor out (b2 + 1).
Example 3: Factoring Out a Common Binomial Factor
Factor each expression.
C. 4z(z2 – 7) + 9(2z3 + 1)
There are no common
– 7) +
+ 1)
factors.
The expression cannot be factored.
4z(z2
9(2z3
Check It Out! Example 3
Factor each expression.
a. 4s(s + 6) – 5(s + 6)
4s(s + 6) – 5(s + 6)
(4s – 5)(s + 6)
The terms have a common
binomial factor of (s + 6).
Factor out (s + 6).
b. 7x(2x + 3) + (2x + 3)
7x(2x + 3) + (2x + 3)
The terms have a common
binomial factor of (2x + 3).
7x(2x + 3) + 1(2x + 3) (2x + 3) = 1(2x + 3)
(2x + 3)(7x + 1)
Factor out (2x + 3).
Check It Out! Example 3 : Continued
Factor each expression.
c. 3x(y + 4) – 2y(x + 4)
3x(y + 4) – 2y(x + 4)
There are no common
factors.
The expression cannot be factored.
d. 5x(5x – 2) – 2(5x – 2)
5x(5x – 2) – 2(5x – 2)
(5x – 2)(5x – 2)
(5x – 2)2
The terms have a common
binomial factor of (5x – 2 ).
(5x – 2)(5x – 2) = (5x – 2)2
Example 4A: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
6h4 – 4h3 + 12h – 8
(6h4 – 4h3) + (12h – 8) Group terms that have a common
number or variable as a factor.
2h3(3h – 2) + 4(3h – 2) Factor out the GCF of each
group.
2h3(3h – 2) + 4(3h – 2) (3h – 2) is another common
factor.
(3h – 2)(2h3 + 4)
Factor out (3h – 2).
Example 4A Continued
Factor each polynomial by grouping.
Check your answer.
Check (3h – 2)(2h3 + 4)
Multiply to check your
solution.
3h(2h3) + 3h(4) – 2(2h3) – 2(4)
6h4 + 12h – 4h3 – 8
6h4 – 4h3 + 12h – 8
The product is the original
polynomial.
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
6b3 + 8b2 + 9b + 12
4r3 + 24r + r2 + 6
2x3 – 12x2 + 18 – 3x
2x3 – 12x2 + 18 – 3x
Examples:
x3  2 x 2  x  2
The GCF of
x 3  2 x 2 is x 2 .
x3  2 x 2  x  2
x 2  x  2  1 x  2 
The GCF of
 x  2 is  1.
  x  2   x  1
2
These two terms must be the
same.
Try These:
Factor by grouping.
a. 8 x 3  2 x 2  12 x  3
b. 4 x 3  6 x 2  6 x  9
c. x 3  x 2  x  1
d. 3a  6b  5a  10ab
2
a. 8 x  2 x  12 x  3
3
The GCF of
8 x3  2 x 2 is 2x 2 .
2
8 x  2 x 12 x  3
3
2
2 x  4 x  1 3  4 x  1
The GCF of
12 x  3 is 3.
2
  4 x  1  2 x  3
2
BACK
b. 4 x  6 x  6 x  9
3
The GCF of
4 x 3  6 x 2 is 2x 2 .
2
4x  6x
3
2
6 x  9
2 x  2 x  3 3  2 x  3
The GCF of
6 x  9 is  3.
2
When you factor a negative out
of a positive, you will get a
negative.
  2 x  3  2 x  3
2
BACK
c. x  x  x  1
3
The GCF of
x 3  x 2 is x 2 .
2
x3  x 2  x  1
The GCF of
x 2  x  1 1 x  1
  x  1  x  1
2
 x  1 is  1.
Now factor the difference of squares.
BACK
  x  1 x  1 x  1
d. 3a  6b  5a  10ab
2
The GCF of
3a  6b is 3.
3a  6b 5a 2  10ab
3  a  2b  5a  a  2b 
The GCF of
5a 2  10ab is 5a.
  a  2b  3  5a 
BACK
Lesson Quiz: Part II
Factor each polynomial by grouping. Check your
answer.
5. 2x3 + x2 – 6x – 3
(2x + 1)(x2 – 3)
6. 7p4 – 2p3 + 63p – 18
(7p – 2)(p3 + 9)
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)

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