1. MATHEMATIC
FACTORIZATION

2. FACTORS

Factors are the numbers you multiply together
to get another number.

The factors of an expression are expressions
that you multiply together to get another
expression.

Factoring is rewriting an expression as a
product of its factors.

3. FACTORING
Whole numbers
3

4. FACTORS
•
•
Factors are the numbers you multiply
together to get a product.
The product 48 has many factors.
48 = 1 x 48
48 = 2 x 24
48 = 3 x 16
48 = 4 x 12
48 = 6 x 8
4

5. Factoring
•
•
Factoring is the process of finding all
the factors of a term.
It is like "splitting" an expression into a
multiplication of simpler expressions
5

6. Factoring Polynomials
Because polynomials come in many shapes
and sizes, there are several patterns you need
to recognize and there are different methods
for solving them.

7. Factoring

To factor a polynomial means to transform it to a
product of two or more factors (usually binomials)

Factoring is the reverse process of FOIL (Double
Distribution)

FOIL can be used to check your work
7

8. Factor each expression
x + 9 x + 20
2

9. To Factor a Polynomial of
2
the Form x ± bx + c
(x
)(x
)
1.
What are factors of c that add up to b?
2.
Set up factors
3.
Plug in the numbers
4.
Check

10. Remember
When the third term is positive in a quadratic trinomial,
the binomial factors have the same sign

11. Third Term is Negative

When the third term is negative the binomial factors have
opposite signs
x ± bx − c = ( x +
2
)( x− )

12. Factoring Polynomials:
Type 1:
Quadratic Trinomials with a Leading coefficient of 1
x − 6x + 9
2
In Grade ten, you learned that there was a pattern with these types of
expressions. When this expression is factored into two binomials, the two
numbers will have a product of 9 and a sum of -6.
x − 6x + 9
2
=
( x - 3 )( x - 3)
2 Binomial Factors

13. Factoring Quadratic Equations with a =
1
Try these examples:
1) x2 + 7x + 12
2) x2 + 8x + 12
3) x2 + 2x – 3
4) x2 – 6x + 8
5) x2 + x – 12
6) x2 – 3x – 10
7) x2 – 8x + 15
8) x2 – 3x – 18
9) x2 – 3x + 2
10) x2 – 10x + 21

14. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
5 x − 19 x − 4
2
There are a variety of ways that you can factor these types
of Trinomials:
a) Factoring by Decomposition
b) Factoring using Temporary Factors
c) Factoring using the Window Pane Method

15. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
5 x − 19 x − 4
2
a) Factoring by Decomposition
1.
2.
Multiply a and c
Look for two numbers that multiply to that product and add to b
3.
Break down the middle term into two terms using those two numbers
4.
Find the common factor for the first pair and factor it out & then find
the common factor for the second pair and factor it out.
5.
From the two new terms, place the common factor in one bracket
and the factored out factors in the other bracket.

16. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
a) Factoring by Decomposition
1.
2.
Multiply a and c
Look for two numbers that multiply
to that product and add to b
3.
Break down the middle term into two
terms using those two numbers
4.
Find the common factor for the first
pair and factor it out & then find the
common factor for the second pair
and factor it out.
5.
From the two new terms, place the
common factor in one bracket and the
factored out factors in the other
bracket.
a × c = −20
The 2 nos. are -20 & 1
5 x − 19 x − 4
2
= 5 x − 20 x + 1x − 4
= 5 x( x − 4) + 1( x − 4)
= ( x − 4)(5 x + 1)
2

17. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
5 x − 19 x − 4
2
b) Factoring using Temporary Factors
1.
2.
Multiply a and c
Look for two numbers that multiply to that product and add to b
3.
4.
Use those numbers as temporary factors.
Divide each of the number terms by a and reduce.
5.
Multiply one bracket by its denominator

18. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
a) Factoring by Temporary Factors
a × c = −20
1. Multiply a and c
2. Look for two numbers that multiply to
that product and add to b
3.Use those numbers as temporary
factors.
4.Divide each of the number terms by a
and reduce.
5.Multiply one bracket by its
denominator
The 2 nos. are -20 & 1
5 x − 19 x − 4
2
= ( x − 20)( x + 1)
20 
1

=  x −  x + 
5 
5

= ( x − 4)(5 x + 1)

19. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
5 x − 19 x − 4
2
c) Factoring using the Window Pane Method
1.
2.
Multiply a and c
Look for two numbers that multiply to that product and add to b
3.
Draw a Windowpane with four panes. Put the first term in the top
left pane and the third term in the bottom right pane.
4.
Use the two numbers for two x-terms that you put in the other two
panes
5.
Take the common factor out of each row using the sign of the first
pane. Take the common factor out of each column using the sign of
the top pane. These are your factors.

20. Factoring Polynomials:
Type 2:
Quadratic Trinomials with a Leading coefficient = 1
a) Factoring by Temporary Factors
a × c = −20
1. Multiply a and c Look for two numbers that
multiply to that product and add to b
The 2 nos. are -20 & 1
5 x − 19 x − 4
2. Draw a Windowpane with four panes. Put
the first term in the top left pane and the
third term in the bottom right pane.
2
x
3. Use the two numbers for two x-terms that
you put in the other two panes.
4.Take the common factor out of each row
using the sign of the first pane. Take the
common factor out of each column using
the sign of the top pane. These are your
factors.
5x
+1
−4
5x 2 − 20 x
+ 1x − 4
= ( x − 4)(5 x + 1)

21. I hope you had a great day with
factorization. The next page provide
all the information where combiled
by ME.

22. Reference List

http://www.slideshare.net/AjarnNoah/factorization12664260?qid=2400a5c7-7c99-450a-821255d342858e82&v=default&b=&from_search=2

http://www.slideshare.net/therockinginventor/maths26473460

http://old.mathleague.com/help/decwholeexp/fractio
n.GIF

https://encrypted-tbn1.gstatic.com/images?
q=tbn:ANd9GcRVRxeaqrAz20YzvJL9FCAMJUN3f2Rxb9GW7SylQxqT2Qr4Pic
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