Factoring Polynomials Factor
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Chapter 13
¡ì 13.1
Factoring
Polynomials
The Greatest Common
Factor
Chapter Sections
Factors
13.1 ¨C The Greatest Common Factor
13.2 ¨C Factoring Trinomials of the Form
x2
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 ¨C writing a polynomial as a product of
polynomials.
+ bx + c
13.3 ¨C Factoring Trinomials of the Form ax2 + bx + c
13.4 ¨C Factoring Trinomials of the Form x2 + bx + c
by Grouping
13.5 ¨C Factoring Perfect Square Trinomials and
Difference of Two Squares
13.6 ¨C Solving Quadratic Equations by Factoring
13.7 ¨C Quadratic Equations and Problem Solving
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Greatest Common Factor
Greatest Common Factor
Example
Greatest common factor ¨C largest quantity that is a
factor of all the integers or polynomials involved.
Find the GCF of each list of numbers.
1) 6, 8 and 46
6=2¡¤3
8=2¡¤2¡¤2
46 = 2 ¡¤ 23
So the GCF is 2.
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.
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
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Greatest Common Factor
Example
Example
Find the GCF of each list of numbers.
1) 12 and 8
12 = 2 ¡¤ 2 ¡¤ 3
8=2¡¤2¡¤2
So the GCF is 2 ¡¤ 2 = 4.
2) 7 and 20
7=1¡¤7
20 = 2 ¡¤ 2 ¡¤ 5
There are no common prime factors so the
GCF is 1.
Martin-Gay, Developmental Mathematics
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
Factoring out the GCF
Example
Example
Factor out the GCF in each of the following
polynomials.
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
1) 6x3 ¨C 9x2 + 12x =
3 ¡¤ x ¡¤ 2 ¡¤ x2 ¨C 3 ¡¤ x ¡¤ 3 ¡¤ x + 3 ¡¤ x ¡¤ 4 =
3x(2x2 ¨C 3x + 4)
2) 14x3y + 7x2y ¨C 7xy =
7 ¡¤ x ¡¤ y ¡¤ 2 ¡¤ x2 + 7 ¡¤ x ¡¤ y ¡¤ x ¨C 7 ¡¤ x ¡¤ y ¡¤ 1 =
7xy(2x2 + x ¨C 1)
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
11
Factoring out the GCF
Example
The first step in factoring a polynomial is to
find the GCF of all its terms.
Factor out the GCF in each of the following
polynomials.
Then we write the polynomial as a product by
factoring out the GCF from all the terms.
1) 6(x + 2) ¨C y(x + 2) =
6 ¡¤ (x + 2) ¨C y ¡¤ (x + 2) =
(x + 2)(6 ¨C y)
2) xy(y + 1) ¨C (y + 1) =
xy ¡¤ (y + 1) ¨C 1 ¡¤ (y + 1) =
(y + 1)(xy ¨C 1)
The remaining factors in each term will form a
polynomial.
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Factoring
Factoring Trinomials
Remember that factoring out the GCF from the terms of
a polynomial should always be the first step in factoring
a polynomial.
Recall by using the FOIL method that
F
I
L
x2
(x + 2)(x + 4) = + 4x + 2x + 8
= x2 + 6x + 8
This will usually be followed by additional steps in the
process.
To factor x2 + bx + c into (x + one #)(x + another #),
note that b is the sum of the two numbers and c is the
product of the two numbers.
Example
Factor 90 + 15y2 ¨C 18x ¨C 3xy2.
90 + 15y2 ¨C 18x ¨C 3xy2 = 3(30 + 5y2 ¨C 6x ¨C xy2) =
3(5 ¡¤ 6 + 5 ¡¤ y2 ¨C 6 ¡¤ x ¨C x ¡¤ y2) =
3(5(6 + y2) ¨C x (6 + y2)) =
3(6 + y2)(5 ¨C x)
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So we¡¯ll be looking for 2 numbers whose product is
c and whose sum is b.
Note: there are fewer choices for the product, so
that¡¯s why we start there first.
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Factoring Polynomials
Example
¡ì 13.2
Factoring Trinomials of the
Form x2 + bx + c
Factor the polynomial x2 + 13x + 30.
Since our two numbers must have a product of 30 and a
sum of 13, the two numbers must both be positive.
Positive factors of 30
Sum of Factors
1, 30
31
2, 15
17
3, 10
13
Note, there are other factors, but once we find a pair
that works, we do not have to continue searching.
So x2 + 13x + 30 = (x + 3)(x + 10).
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Factoring Polynomials
Prime Polynomials
Example
Example
Factor the polynomial x2 ¨C 11x + 24.
Since our two numbers must have a product of 24 and a
sum of -11, the two numbers must both be negative.
Negative factors of 24
Sum of Factors
¨C 1, ¨C 24
¨C 25
¨C 2, ¨C 12
¨C 14
¨C 3, ¨C 8
¨C 11
Factor the polynomial x2 ¨C 6x + 10.
Since our two numbers must have a product of 10 and a
sum of ¨C 6, the two numbers will have to both be negative.
Negative factors of 10
Sum of Factors
¨C 1, ¨C 10
¨C 11
¨C 2, ¨C 5
¨C7
Since there is not a factor pair whose sum is ¨C 6,
x2 ¨C 6x +10 is not factorable and we call it a prime
polynomial.
So x2 ¨C 11x + 24 = (x ¨C 3)(x ¨C 8).
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Factoring Polynomials
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Check Your Result!
Example
Factor the polynomial x2 ¨C 2x ¨C 35.
Since our two numbers must have a product of ¨C 35 and a
sum of ¨C 2, the two numbers will have to have different signs.
Factors of ¨C 35
Sum of Factors
¨C 1, 35
34
1, ¨C 35
¨C 34
¨C 5, 7
2
5, ¨C 7
¨C2
You should always check your factoring
results by multiplying the factored polynomial
to verify that it is equal to the original
polynomial.
Many times you can detect computational
errors or errors in the signs of your numbers
by checking your results.
So x2 ¨C 2x ¨C 35 = (x + 5)(x ¨C 7).
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