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Rules of Factoring Polynomials
A presentation for RHHS Grade 10
Wonderful Students
By Ms. Wang
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Rules
Step by Step
Easy Problems
x 2  14x  24
3
2
3x  9 x
2
4s  16
Hard Problems
6 x3  15x 2  9 x
8
2 x  32
Medium Problems
4my  20m  3 py 15p
8s 2  200t 2
6 y2  5y  6
Word Problems
Division of polynomial by monomial
Find dimensions when area is given
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Rules of Factoring
Flowchart of Factoring polynomials
GCF and Leading “-”
Factor out GCF and rewrite the left polynomial
inside a parenthesis
Binomial
a b
2
2
Difference of two
squares
(a  b)(a  b)
Trinomial
ax  b
cx  d
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(ax  b)(cx  d )
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Step by Step
 Is there a GCF?
Yes
 Factor as the product of the GCF and one other
factor—i.e. GCF•(the other factor). Look at the other
factor and go to the next step below with it.
No
 Go the the next step.
 Is it a binomial?
Yes
 Is it a difference of two squares? (a2-b2)
• Yes—Factor as (a+b)(a-b).
• No—It can’t be factored any more.
No
 Go to the next step.
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Is it a trinomial?
Yes
Use the X BOX pattern to look for factors.
No
Go to the next step.
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NOTE:
At EVERY step along the way, you
must look at the factors that you get to
see if they can be factored any more.
Factoring completely means that no factors can be
broken down any further using any of the rules
you’ve learned.
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Practice
Factor completely.
x  14x  24
2
No.
Is there a GCF?
Is it a binomial or trinomial?
It’s a trinomial.
XBOX
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x2
24
x
12
x
2
Use your handy-dandy calculator or
your super math skills to find 12
and 2 as the factors to use.
12x + 2x = 14x
Rewrite the equation with those two factors in the middle.
x  14x  24
2
 ( x  12)(x  2)
Write the two factors.
Neither one of these factors can be
broken down any more, so you’re done.
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Factor completely
3x  9 x
3
2
Is there a GCF?
Yes. Write the GCF first and the remaining factor after it.
 3x ( x  3)
2
Look at the remaining factor. (x-3)
Is it a binomial or trinomial?
It’s a binomial. Is it a difference of two squares? (a2-b2)
No. You can’t do anything else.
 3x ( x  3)
2
is the completely factored form.
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Factor completely
4s  16
2
Is there a GCF?
Yes. Write the GCF first and the remaining factor after it.
 4(s  4) Look at the remaining factor. (s -4)
2
2
Is it a binomial, trinomial?
It’s a binomial. Is it a difference of two squares? (a2-b2)
Yes. s2 is a square (s • s) and 4 is a square (2 • 2).
Factor as (s+2)(s-2). Then write the complete
factorization.
 4( s  2)(s  2)
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Factor completely
6y  5y  6
2
No.
Is there a GCF?
Is it a binomial or trinomial?
It’s a trinomial.
6y2
2y
b
-3
3y
2
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Check the sum of the cross product
is -5y
2*2y + 3y *(-3)
= -5y
Rewrite the equation with those two factors in the middle.
6y  5y  6
2
 (2 y  3)(3 y  2)
Write the two factors.
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Factor completely
6 x  15x  9 x
3
2
Is there a GCF?
Yes. Write the GCF first and the remaining factor after it.
 3x(2x  5x  3) Look at the remaining factor.
2
(2x2  5x  3)
Is it a binomial or trinomial?
It’s a trinomial.
2x2
2x
x
-3
-1
3
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Check the sum of the cross product
is 5x.
2x*3 + x*(-1)
=5x
Rewrite the equation with those two factors in the middle.
3x(2 x  5x  3)
2
 3x( x  3)(2 x  1)
Write all three factors.
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Factor completely
2 x  32
8
Is there a GCF?
Yes. Write the GCF first and the remaining factor after it.
2( x 16)
8
Look at the remaining factor.
( x8  16)
Is it a binomial, trinomial, or four-term polynomial?
It’s a binomial. Is it a difference of two squares? (a2-b2)
Yes. x8 is a square (x4 • x4) and 16 is a square
(4 • 4). Factor as (x4 + 4)(x4 - 4).
So far we have 2(x4 + 4)(x4 - 4).
(Please continue—not done yet!!)
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2(x4 +4)(x4 -4)
Look at what you have. Can either of the binomials be broken
down?
(x4 +4)
Is this binomial a difference of two squares? (a2-b2)
No. It can’t be broken down. So, we have to keep this factor.
What about the other binomial?
(x4 -4)
Is this binomial a difference of two squares? (a2-b2)
Yes. x4 is a square (x2 • x2) and 4 is a square
(2 • 2). Factor as (x2 + 2)(x2 - 2).
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Put it all together.
2 x  32
8
 2( x  16)
8
=2(x4 +4)(x4 -4)
=2(x4 +4)(x2 +2)(x2 -2)
Not a difference of
squares. Can’t go
any farther!!
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Word Problem #1
What is the quotient when
12x3  8x 2  16x is divided by 4x?
This question is
3
2
3
2
12
x

8
x

16
x
12
x
8
x
16x
asking you to find



4x
4x
4x 4x
the OTHER
FACTOR after you
take out the greatest
2
3x
 2x  4
common factor of 4x.
Simplify each term.
3x  2 x  4
2
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Word Problem #2
A rectangular garden plot has an area
represented by the expression
18x 2  3x  28
Find the dimensions of the garden plot.
This is a factoring problem. You need to
find the two factors that multiply together
to give you 18x 2  3x  28
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18x  3x  28
2
No.
Is there a GCF?
Is it a binomial, trinomial, or four-term polynomial?
It’s a trinomial.
18x2
3x
6x
-28
-4
7
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Check the sum of the cross product
of the XBOX is -3x.
Rewrite the equation with those two factors in the middle.
18x  3x  28
2
 (3x  4)(6 x  7)
Write the two factors.
Length is 3x - 4 and width is 6x + 7
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The End
Practice Makes Master!