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Factoring by GCF
Factoring
Put the expression in a division tower
Continue to divide by numbers or
variables until there is no number or
variable common to all terms.
Put the numbers and variables along
the side on the outside of the
parentheses.
Put the top expression on the inside of
parentheses.
Example
1
4
3
2
56x – 32x – 72x
Example 2: Factoring Out a
Common Binomial Factor
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) 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 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.
2(3h – 2)(h3 + 2)
Factor out (3h – 2).
Example 4: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
(5y4 – 15y3) + (y2 – 3y)
Group terms.
5y3(y – 3) + y(y – 3)
Factor out the GCF of
each group.
5y3(y – 3) + y(y – 3)
(y – 3) is a common factor.
y(y – 3)(5y2 + 1)
Factor out (y – 3).
Example 5: Factoring with
Opposites
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.
Write (6 – x) as –1(x – 6).
Simplify. (x – 6) is a
common factor.
Factor out (x – 6).
Example 6
Factor each polynomial. Check your answer.
15x2 – 10x3 + 8x – 12
(15x2 – 10x3) + (8x – 12)
5x2(3 – 2x) + 4(2x – 3)
Group terms.
Factor out the GCF of
each group.
5x2(3 – 2x) + 4(–1)(3 – 2x) Write (2x – 3) as –1(3 – 2x).
5x2(3 – 2x) – 4(3 – 2x)
-1(2x - 3)(5x2 – 4)
Simplify. (3 – 2x) is a
common factor.
Factor out (3 – 2x).
Try these…
Factor each polynomial. Check your answer.
1. 16x + 20x3
4x(4 + 5x2)
2. 4m4 – 12m2 + 8m
4m(m3 – 3m + 2)
Factor each expression.
3. 7k(k – 3) + 4(k – 3)
4. 3y(2y + 3) – 5(2y + 3)
(k – 3)(7k + 4)
(2y + 3)(3y – 5)
Try these (cont)…
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)