Transcript Document

Chapter 13
Factoring
Polynomials
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
13.1
The Greatest Common
Factor
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Factoring Polynomials
When an integer is written as a product of integers,
each of the integers in the product is a factor of the
original number. The product is the factored form of
the integer.
When a polynomial is written as a product of
polynomials, each of the polynomials in the product is
a factor of the original polynomial. The product is the
factored form of the polynomial.
The process of writing a polynomial as a product is
called factoring the polynomial.
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Greatest Common Factor
Greatest common factor – largest quantity that is a
factor of all the integers or polynomials involved.
Finding the GCF of a List of Integers or Terms
1. Write each number or polynomial as a product of
prime factors.
2. Identify common prime factors.
3. Take the product of all common prime factors.
• If there are no common prime factors, GCF is 1.
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Example
Find the GCF of each list of numbers.
a. 12 and 8
12 = 2 · 2 · 3
8=2·2·2
So the GCF is 2 · 2 = 4.
b. 7 and 20
7=1·7
20 = 2 · 2 · 5
There are no common prime factors so the
GCF is 1.
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Example
Find the GCF of each list of numbers.
a. 6, 8 and 46
6=2·3
8=2·2·2
46 = 2 · 23
So the GCF is 2.
b. 144, 256 and 300
144 = 2 · 2 · 2 · 2 · 3 · 3
256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2
300 = 2 · 2 · 3 · 5 · 5
So the GCF is 2 · 2 = 4.
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Example
Find the GCF of each list of terms.
a. x3 and x7
x3 = x · x · x
x7 = x · x · x · x · x · x · x
So the GCF is x · x · x = x3
b. 6x5 and 4x3
6x5 = 2 · 3 · x · x · x
4x3 = 2 · 2 · x · x · x
So the GCF is 2 · x · x · x = 2x3
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Example
Find the GCF of the following list of terms.
a3b2, a2b5 and a4b7
a3b2 = a · a · a · b · b
a2b5 = a · a · b · b · b · b · b
a4b7 = a · a · a · a · b · b · b · b · b · b · b
So the GCF is a · a · b · b = a2b2
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Helpful Hint
Remember that the GCF of a list of terms
contains the smallest exponent on each
common variable.
smallest exponent on x
smallest exponent on y
The GCF of x3y5, x6y4, and x4y6is x3y4.
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Factoring Polynomials
The first step in factoring a polynomial is to
find the GCF of all its terms.
Then we write the polynomial as a product by
factoring out the GCF from all the terms.
The remaining factors in each term will form a
polynomial.
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Example
Factor out the GCF in each of the following
polynomials.
a. 6x3 – 9x2 + 12x =
3 · x · 2 · x2 – 3 · x · 3 · x + 3 · x · 4 =
3x(2x2 – 3x + 4)
b. 14x3y + 7x2y – 7xy =
7 · x · y · 2 · x2 + 7 · x · y · x – 7 · x · y · 1 =
7xy(2x2 + x – 1)
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Example
Factor out the GCF in each of the following
polynomials.
a. 6(x + 2) – y(x + 2) =
6 · (x + 2) – y · (x + 2) =
(x + 2)(6 – y)
b. xy(y + 1) – (y + 1) =
xy · (y + 1) – 1 · (y + 1) =
(y + 1)(xy – 1)
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Factoring
Remember that factoring out the GCF from the terms of
a polynomial should always be the first step in factoring
a polynomial.
This will usually be followed by additional steps in the
process.
Example:
Factor 90 + 15y2 – 18x – 3xy2.
90 + 15y2 – 18x – 3xy2 = 3(30 + 5y2 – 6x – xy2)
= 3(5 · 6 + 5 · y2 – 6 · x – x · y2)
= 3(5(6 + y2) – x (6 + y2))
= 3(6 + y2)(5 – x)
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Factoring by Grouping
Step 1: Group the terms in two groups so that
each group has a common factor.
Step 2: Factor out the GCF from each group.
Step 3: If there is a common binomial factor,
factor it out.
Step 4: If not, rearrange the terms and try
these steps again.
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Example
Factor by grouping.
21x3y2 – 9x2y + 14xy – 6
= (21x3y2 – 9x2y) + (14xy – 6)
= 3x2y(7xy – 3) + 2(7xy – 3)
= (7xy – 3)(3x2 + 2)
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