Factoring Polynomials

1
Factoring Polynomials
Chapter 6
2
6.1

• The Greatest Common Factor and Factoring by
  Grouping

3
Factors

• Factors (either numbers or 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.
• When a polynomial is written as a product of
  polynomials, each of the polynomials in the
  product is a factor of the original polynomial.
• Factoring writing a polynomial as a product of
  polynomials

4
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
• Write each number as a product of prime numbers.
• Identify the common prime factors.
• The product of all common prime factors found in
  step 2 is the greatest common factor. If there
  are no common prime factors, the greatest common
  factor is 1.

5
Greatest Common Factor
Example
Find the GCF of each list of numbers.

• 12 and 8
• 12 2 2 3
• 8 2 2 2
• So the GCF is 2 2 4.
• 7 and 20
• 7 1 7
• 20 2 2 5
• There are no common prime factors so the GCF is 1.

6
Greatest Common Factor
Example
Find the GCF of each list of numbers.

• 6, 8 and 46
• 6 2 3
• 8 2 2 2
• 46 2 23
• So the GCF is 2.
• 144, 256 and 300
• 144 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.

7
Greatest Common Factor
Example
Find the GCF of each list of terms.

• x3 and x7
• x3 x x x
• x7 x x x x x x x
• So the GCF is x x x x3
• 6x5 and 4x3
• 6x5 2 3 x x x x x
• 4x3 2 2 x x x
• So the GCF is 2 x x x 2x3

8
Greatest Common Factor
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

Notice that the GCF of terms containing variables
will use the smallest exponent found amongst the
individual terms for each variable.
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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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Factoring out the GCF
Example
Factor out the GCF in each of the following
polynomials.
1) 6x3 9x2 12x 3 x 2 x2 3 x 3
x 3 x 4 3x(2x2 3x 4) 2) 14x3y
7x2y 7xy 7 x y 2 x2 7 x y x
7 x y 1 7xy(2x2 x 1)
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Factoring out the GCF
Example
Factor out the GCF in each of the following
polynomials.

• 1) 6(x 2) y(x 2)
• 6 (x 2) y (x 2)
• (x 2)(6 y)
• 2) xy(y 1) (y 1)
• xy (y 1) 1 (y 1)
• (y 1)(xy 1)

12
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)

13
Factoring by Grouping
Factoring polynomials often involves additional
techniques after initially factoring out the
GCF. One technique is factoring by grouping.
Example

• Factor xy y 2x 2 by grouping.
• Notice that, although 1 is the GCF for all four
  terms of the polynomial, the first 2 terms have a
  GCF of y and the last 2 terms have a GCF of 2.
• xy y 2x 2 x y 1 y 2 x 2 1
• y(x 1) 2(x 1) (x 1)(y 2)

14
Factoring by Grouping

• To Factor a Four-Term Polynomial by Grouping
• Group the terms in two groups of two terms so
  that each group has a common factor.
• Factor out the GFC from each group.
• If there is now a common binomial factor in the
  groups, factor it out.
• If not, rearrange the terms and try these steps
  again.

15
Factoring by Grouping
Example
Factor each of the following polynomials by
grouping.

• 1) x3 4x x2 4 x x2 x 4 1 x2
  1 4
• x(x2 4) 1(x2 4)
• (x2 4)(x 1)
• 2) 2x3 x2 10x 5 x2 2x x2 1 5
  2x 5 ( 1)
• x2(2x 1) 5(2x 1)
• (2x 1)(x2 5)

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Factoring by Grouping
Example

• Factor 2x 9y 18 xy by grouping.
• Neither pair has a common factor (other than 1).
• So, rearrange the order of the factors.
• 2x 18 9y xy 2 x 2 9 9 y x y
  
• 2(x 9) y(9 x)
• 2(x 9) y(x 9) (make sure the factors
  are identical)
• (x 9)(2 y)