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.
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
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
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.
We are going to start today with the second easiest form of factoring (the easiest being factoring out the GCF).
All of the polynomials that we will be factoring today will be quadratic (or quadratic form), mostly trinomials. Some, towards the end, aren’t polynomials, but we will still be able to factor them. Also, most of them will start with x2. There will be some special cases where the x2 has a coefficient other than one, but we are going to save the bulk of that type until day 2 of factoring.
Factoring mainly requires an understanding of how polynomial multiplication works and logic. We are basically going to reason our way through this process.
Here is the actual procedure we are going to use to factor quadratic trinomials of the form x2 + bx + c.
This is going to work for us because we know what happens when we multiply two linear binomials.
The x2 term comes from x*x. These x’s come from the first thing we multiply together. That means that our factors have to start (x …) (x …).
The numbers that go after the x have to multiply together to give us “c”, which is why we need to find factors of “c”. We are going to include the sign when we think of factors. It will make a difference!
We also know that the “bx” term comes from, in essence, adding the factors of “c”. So, if c has multiple factor pairs, we need to find the one pair that will add to b. Remember to incorporate the sign of the factors of c.
The final step is to always check to make sure everything works right. It is really easy to pick the wrong factors or the wrong sign on the factors.
Examples.
I’ll go through the reasoning on some of these, but give the solution to all of them.
#1 a) We need to look for factors of +12 that add to 7. This means they both have to be positive. The possible factor pairs are (1, 12), (2, 6), and (3,4).
b) Which of these pairs add up to 7? That would be 3 and 4.
c) It seems that (x + 3)(x + 4) might be the factors. They add up to 7 and multiply out to 12.
d) Time to check (x+3)(x+4) = x2+ 4x + 3x + 12 = x2 + 7x + 12!
#2. (x + 6)(x+2)
#3a) We need to look for factors of –3 that add to 2. This means that one is negative and one is positive. Also that the positive one is “bigger.” The possible factor pairs are (1, -3) and (-1, 3).
b) Which of these add up to +2? (-1, 3).
c) (x – 1) (x+3) looks to be the answer.
d) Time to check (x-1)(x+3) = x2 + 3x – x – 3 = x2 + 2x – 3
#4a) We need to look for factors of +8 that add to –6. This means that they both are negative. The possible factor pairs are (-1, -8), (-2, -4).
b) Which of these add up to –6? (-2, -4)
c) (x – 2)(x –4) might be the solution.
d) Check: (x-2)(x-4) = x2 – 4x – 2x + 8 = x2 – 6x + 8
#5. ( x + 4)(x – 3)
#6) (x-5)(x+2)
#7) (x – 3)(x-5)
#8) ( x – 6)(x + 3)
#9) ( x – 2)(x-1)
#10) (x – 7)(x-3)
