Factor polynomials by various methods

1. 0.4 Factoring Polynomials
Chapter 0 Review of Basic Concepts

2. Concepts and Objectives
⚫ Factoring out the greatest common factor
⚫ Factoring by Grouping
⚫ Factoring Trinomials
⚫ Factoring by Substitution

3. Factoring Polynomials
⚫ The process of finding polynomials whose product
equals a given polynomial is called factoring.
⚫ For example, since 4x + 12 = 4(x + 3), both 4 and x + 3 are
called factors of 4x + 12.
⚫ A polynomial that cannot be written as a product of two
polynomials of lower degree is a prime polynomial.
⚫ One nice aspect of this process is that it has a built-in
check: whatever factors you come up with, you should
be able to multiply them and get your starting
expression.

4. Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
⚫
⚫
⚫
5 2
9y y+
2
6 8 12x t xt t+ +
( ) ( ) ( )
3 2
14 1 28 1 7 1m m m+ − + − +

5. Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
⚫ GCF: y2
⚫ GCF: 2t
⚫
GCF: 7m + 1
5 2
9y y+
2
6 8 12x t xt t+ +
( ) ( ) ( )
3 2
14 1 28 1 7 1m m m+ − + − +
( )32
9 1y y +
( )2
62 3 4xt x+ +
( ) ( ) ( )
2
47 1 2 1 1 1m mm  + −+ − +
 

6. Factoring Out the GCF (cont.)
We can clean up that last problem just a little more:
( ) ( ) ( )
( ) ( ) ( )
( )
( )( )
 + + − + −
 
 + + + − + − 
 + + + − − − 
+ −
2
2
2
2
7 1 2 1 4 1 1
7 1 2 2 1 4 1 1
7 1 2 4 2 4 4 1
7 1 2 3
m m m
m m m m
m m m m
m m

7. Factoring by Grouping
⚫ When a polynomial has more than three terms, it can
sometimes be factored using factoring by grouping.
⚫ For example, to factor
ax + ay + 6x + 6y,
group the terms so that each group has a common factor.
( ) ( )
( ) ( )
( )( )
6 6 6 6
6
6
ax ay x y ax ay x y
a x y x y
x y a
+ + + = + + +
= + + +
= + +
x + y is the GCF
of the expression
above.

8. Factoring by Grouping (cont.)
Examples: Factor each polynomial by grouping.
⚫
⚫
2 2
7 3 21mp m p+ + +
2 2
2 2y az z ay+ − −

9. Factoring by Grouping (cont.)
Examples: Factor each polynomial by grouping.
⚫
⚫
( ) ( )
( )( )
2 2 2 2
2
7 3 21 7 3 7
7 3
mp m p m p p
p m
+ + + = + + +
= + +
2 2
2 2y az z ay+ − −

10. Factoring by Grouping (cont.)
Examples: Factor each polynomial by grouping.
⚫
⚫
( ) ( )
( )( )
2 2 2 2
2
7 3 21 7 3 7
7 3
mp m p m p p
p m
+ + + = + + +
= + +
( ) ( )
( ) ( )
( )( )
+ − − = − − +
= − + − +
= − − −
= − −
2 2 2 2
2 2
2 2
2
2 2 2 2
2 2
2
2
y az z ay y z ay az
y z ay az
y z a y z
y z a
You can rearrange
the terms to make
grouping easier.

11. Factoring by Grouping (cont.)
Examples: Factor each polynomial by grouping.
⚫ 3 2
4 2 2 1x x x+ − −

12. Factoring by Grouping (cont.)
Examples: Factor each polynomial by grouping.
⚫ ( ) ( )
( ) ( )
( )( )
3 2 3 2
2
2
4 2 2 1 4 2 2 1
2 2 1 1 2 1
2 1 2 1
x x x x x x
x x x
x x
−
+
+ − − = +
= −
= +
+
+
−

13. Factoring Trinomials
If you have an expression of the form ax2 +bx + c, you can
use one of the following methods to factor it:
⚫ X-method (a = 1): If a = 1, this is the simplest method to
use. Find two numbers that multiply to c and add up
to b. These two numbers will create your factors.
⚫ Example: Factor x2 ‒ 5x ‒ 14.
‒14
‒7 2
‒5
( )( )2
5 14 7 2x x x x− − = − +
c
b

14. Factoring Trinomials (cont.)
⚫ Reverse box: If a is greater than 1, you can use the
previous X method to split the middle term (ac goes on
top) and use either grouping or the box method, and
then find the GCF of each column and row.
⚫ Example: Factor
Now, find the GCF of each line.
− −2
4 5 6y y
‒24
‒8 3
5
4y2 ‒8y
3y ‒6
ac
b

15. Factoring Trinomials (cont.)
⚫ Reverse box: If a is greater than 1, you can use the
previous X method to split the middle term (ac goes on
top) and use either grouping or the box method, and
then find the GCF of each column and row.
⚫ Example: Factor − −2
4 5 6y y
‒24
‒8 3
5
y ‒2
4y 4y2 ‒8y
3 3y ‒6
( )( )− − = + −2
4 5 6 4 3 2y y y y

16. Factoring Trinomials (cont.)
⚫ Grouping: This method is about the same as the Reverse
Box, except that it is not in a graphic format.
⚫ Example: Factor 2
2 6x x− −
‒12
‒4 3
‒1
( ) ( )
( ) ( )
( )( )
2 2
2
2 6 2 6
2 4 3 6
2 2 3 2
2 2 3
4 3x x x
x x x
x x
x
x
x
x x
− − = −
= − + −
=
=
+
−
−
− + −
+

17. Perfect Square Trinomials
⚫ We can use the reverse of the special patterns we saw
last class to quickly factor perfect square trinomials if
we can recognize the pattern.
⚫ If you encounter a trinomial which fits this pattern, you
can quickly factor it by taking the square roots of the
first and last term.
( )
22 2
2a ab b a b + = 

18. Perfect Square Trinomials
⚫ Example: Factor 9x2 ‒ 12x + 4

19. Perfect Square Trinomials
⚫ Example: Factor 9x2 ‒ 12x + 4
The first thing to notice is that the first and last terms
are perfect squares, and that the middle term is two
times the product of the square roots.
To factor this, put the two square roots together, along
with whatever the sign is between the first and second
term.
2
9 3 and 4 2x x= = ( )( )2 3 4 12x x=
( )
22
9 12 4 3 2x x x+ =− −

20. Perfect Square Trinomials
Examples: Factor the following
⚫
⚫
⚫
2 2
16 40 25p pq q− +
2 2
36 84 49x y xy+ +
2
81 90 25a a− +

21. Perfect Square Trinomials
Examples: Factor the following
⚫
⚫
⚫
( ) ( )( ) ( )
( )
2 22 2
2
16 40 25 4 2 4 5 5
4 5
p pq q p p q q
p q
− + = − +
= −
( ) ( )( )
( )
22 2 2
2
36 84 49 6 2 6 7 7
6 7
x y xy xy xy
xy
+ + = + +
= +
( ) ( )( )
( )
22 2
2
81 90 25 9 2 9 5 5
9 5
a a a a
a
− + = − +
= −

22. Factoring Binomials
⚫ If you are asked to factor a binomial (2 terms), check
first for common factors, then check to see if it fits one of
the following patterns:
⚫ Note: There is no factoring pattern for a sum of
squares (a2 + b2) in the real number system.
Difference of Squares a2 ‒ b2 = a + ba ‒ b
Sum/Diff. of Cubes ( )( )3 3 2 2
a b a b a ab b =  +

23. Factoring Binomials (cont.)
Examples
⚫ Factor
⚫ Factor
⚫ Factor
2
4 81x −
3
27x −
3
3 24x +
( )
( )( )
2 2
2 9
2 9 2 9
x
x x
= −
= − +
( )( )
3 3
2
3
3 3 9
x
x x x
=
+
−
= − +
( ) ( )
( )( )
3 3 3
2
3 8 3 2
3 2 2 4
x x
x x x
= + = +
= + − +

24. Factoring Binomials (cont.)
EVERY TIME YOU DO THIS:
A KITTEN DIES
( )
2 2 2
x y x y+ = +
Remember:

25. Factoring By Substitution
Sometimes a polynomial can be more easily factored by
substituting one expression for another.
⚫ Example: Factor
Let u = z2, so the expression becomes .
This can be factored using the method of your choice
to .
Replace the u back with z2, and we get:
4 2
6 13 5z z− −
− −2
6 13 5u u
( )( )2 5 3 1u u− +
( )( )2 2
2 5 3 1z z− +

26. Factoring By Subsituttion (cont.)
⚫ Example: Factor
This time, let , so the expression becomes
10 Using the factoring method of your
choice, this factors to .
Now, we replace the u and clean up the mess:
( ) ( )
2
10 2 1 19 2 1 15a a− − − −
( )2 1u a= −
2
19 15.u u− −
( )( )5 3 2 5u u+ −
( ) ( )5 2 1 3 2 2 1 5a a  − + − −  
( )( )10 5 3 4 2 5a a− + − −
( )( ) ( )( )10 2 4 7 2 5 1 4 7a a a a− − = − −

27. Classwork
⚫ College Algebra
⚫ 0.4 Assignment – page 40: 4-22, pg. 30: 44-76 (×4),
pg. 20: 96-106
⚫ 0.4 Classwork Check
⚫ Quiz 0.3
I’m allowing 2 class days for this section, so the classwork
check and the quiz are not due until next Tuesday (9/1).