Factoring
Polynomials
The Greatest Common
Factor
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.
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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) Prime factor the numbers.
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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Greatest Common Factor
Example
Find the GCF of each list of terms.
1) x3 and x7
x3 = x · x · x
x7 = x · x · x · x · x · x · x
So the GCF is x · x · x = x3
2) 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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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 out the GCF
Example
1) 6x3 – 9x2 + 12x =
3 · x · 2 · x2 – 3 · x · 3 · x + 3 · x · 4 =
3x(2x2 – 3x + 4)
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Factoring out the GCF
Example
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 Trinomials of
2
the Form x + bx + c
by Grouping
Factoring by Grouping
Factoring a Four-Term Polynomial by Grouping
1) Arrange the terms so that the first two terms have a
common factor and the last two terms have a common
factor.
2) For each pair of terms, use the distributive property to
factor out the pair’s greatest common factor.
3) If there is now a common binomial factor, factor it out.
4) If there is no common binomial factor in step 3, begin
again, rearranging the terms differently.
•
If no rearrangement leads to a common binomial
factor, the polynomial cannot be factored.
•
Remember: all have a common factor of 1!
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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
y(x + 1) + 2(x + 1) = (x + 1)(y + 2)
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Factoring by Grouping
Example
Factor the following polynomial by grouping.
2) x3 + 4x + x2 + 4 =
x · x2 + x · 4 + 1 · x2 + 1 · 4 =
x(x2 + 4) + 1(x2 + 4) =
(x2 + 4)(x + 1)
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Factoring by Grouping
Example
Factor the following polynomial by grouping.
3) 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)
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Lab
Textbook
page:
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