Notes: Unit 11
Review/Practice
Warm Up
Factor each trinomial.
1. x2 + 13x + 40 (x + 5)(x + 8)
2. 5x2 – 18x – 8
(5x + 2)(x – 4)
3. Factor the perfect-square trinomial
16x2 + 40x + 25. (4x + 5)(4x + 5)
4. Factor 9x2 – 25y2 using the difference of
two squares. (3x + 5y)(3x – 5y)
I. Factoring Completely
Recall that a polynomial is fully or
completely factored when it is written
as a product of monomials and
polynomials whose terms have no
common factors other than 1.
Ex 1: Tell whether each expression is completely factored. If not, factor it.
A. 3x2(6x – 4)
3x2(6x – 4)
6x2(3x – 2)
6x – 4 can be factored further.
Factor out 2, the GCF of 6x and – 4.
6x2(3x – 2) is completely factored.
B. (x2 + 1)(x – 5)
(x2 + 1)(x – 5)
Neither x2 +1 nor x – 5 can be
factored further.
(x2 + 1)(x – 5) is completely factored.
C. 5x(x2 – 2x – 3)
5x(x2 – 2x – 3)
(x2 – 2x – 3) can be
factored further.
5x(x + 1)(x – 3)
Factor x2 – 2x – 3.
5x(x + 1)(x – 3) is completely factored.
Caution
x2 + 4 is a sum of squares, and cannot be factored.
More Examples
Tell whether each expression is completely factored. If not, factor it.
A. 5x2(x – 1)
5x2(x – 1)
Neither 5x2 nor x – 1 can be factored
further.
5x2(x – 1) is completely factored.
B. (4x + 4)(x + 1)
(4x + 4)(x + 1)
4x + 4 can be further factored.
4(x + 1)(x + 1)
Factor out 4, the GCF of 4x and 4.
4(x + 1)2 is completely factored.
II. Multiple Factoring Techniques
To factor a polynomial completely, you may need to use more
than one factoring method. Use the steps on the next slide to
factor a polynomial completely.
Ex 1: Factor 10x2 + 48x + 32 completely. Check your answer.
10x2 + 48x + 32
2(5x2 + 24x + 16)
Factor out the GCF.
2(5x + 4)(x + 4)
Factor 5x2 + 24x + 16.
Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)
= 10x2 + 40x + 8x + 32
= 10x2 + 48x + 32

Ex 2: Factor 8x6y2 – 18x2y2 completely. Check your answer.
8x6y2 – 18x2y2
2x2y2(4x4 – 9)
2x2y2(2x2 – 3)(2x2 + 3)
Factor out the GCF. 4x4 – 9 is a
perfect-square trinomial of
the form a2 – b2.
a = 2x2, b = 3
Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9)
= 8x6y2 – 18x2y2

Ex 3: Factor each polynomial completely. Check your answer.
4x3 + 16x2 + 16x
4x3 + 16x2 + 16x
4x(x2 + 4x + 4)
4x(x + 2)2
Factor out the GCF. x2 + 4x + 4 is a
perfect-square trinomial of the
form a2 + 2ab + b2.
a = x, b = 2
Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4)
= 4x(x2 + 4x + 4)
= 4x3 + 16x2 + 16x 
Ex 4: Factor each polynomial completely. Check your answer.
2x2y – 2y3
2y(x2 – y2)
Factor out the GCF. 2y(x2 – y2) is a
perfect-square trinomial of the
form a2 – b2.
2y(x + y)(x – y)
a = x, b = y
2x2y – 2y3
Check 2y(x + y)(x – y) = 2y(x2 – xy + xy – y2)
= 2x2y – 2xy2 + 2xy2 – 2y3
= 2x2y – 2y3

Helpful Hint
For a polynomial of the form ax2 + bx + c, if there are no integers whose sum is b
and whose product is ac, then the polynomial is said to be unfactorable, or prime.
Ex 5: Factor each polynomial completely.
9x2 + 3x – 2
9x2+ 3x – 2
(3x – 1)(3x + 2)
Ex 6: Factor each polynomial completely.
12b3 + 48b2 + 48b
12b(b2 + 4b + 4)
12b(b + 2)(b + 2)
12b(b + 2)2
The GCF is 12b; (b2 + 4b + 4)
is a perfect-square
trinomial in the form of a2
+ 2ab + b2.
Ex 7: Factor each polynomial completely.
4y2 + 12y – 72
4(y2 + 3y – 18)
4(y – 3)(y + 6)
Factor out the GCF. There is no
pattern. b = 3 and c = –18;
look for factors of –18 whose
sum is 3.
Ex 8: Factor each polynomial completely.
(x4 – x2)
x2(x2 – 1)
Factor out the GCF.
x2(x + 1)(x – 1)
x2 – 1 is a difference of two
squares.
Ex 9: Factor each polynomial completely.
3x2 + 7x + 4
3x2 + 7x + 4
(3x + 4)(x + 1)
Ex 12: Factor each polynomial completely.
2p5 + 10p4 – 12p3
2p3(p2 + 5p – 6)
2p3(p + 6)(p – 1)
Factor out the GCF.
Ex 13: Factor each polynomial completely.
9q6 + 30q5 + 24q4
3q4(3q2
+ 10q + 8)
3q4(3q + 4)(q + 2)
Factor out the GCF. There is no
pattern.
Ex 14: Factor each polynomial completely.
2x4 + 18
2(x4 + 9)
Factor out the GFC.
x4 + 9 is the sum of squares
and that is not factorable.
2(x4 + 9) is completely factored.
Lesson Quiz
Tell whether the polynomial is completely factored. If not, factor it.
1. (x + 3)(5x + 10)
2. 3x2(x2 + 9)
no; 5(x+ 3)(x + 2)
completely factored
Factor each polynomial completely. Check your answer.
3. x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48
(x + 4)(x2 + 3)
5. 18x2 – 3x – 3
3(3x + 1)(2x – 1)
7. 5x – 20x3 + 7 – 28x2
4(x + 6)(x – 2)
6. 18x2 – 50y2
2(3x + 5y)(3x – 5y)
(1 + 2x)(1 – 2x)(5x + 7)