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Unit 2: Factoring
Part I
MM 218
McGrath
Vocubulary
• A factor is a number, variable, or algebraic expression
multiplying another number, variable or algebraic
expression.
Example: 4*3 (both the “4” and the “3” are factors)
Example: 2x*3y2 (both the “2x” and the “3y2” are
factors)
Vocabulary
• A greatest common factor (GCF) is the largest factor
common to 2 or more terms
Example: Find the GCF of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common Factors: 1, 2, 3, 6
Greatest Common Factor: 6
GCF Cont’d
• Example: Find the GCF of 6x3 and 15x2
Factors of 6 : 1, 2, 3, 6
Factors of x3: x*x*x
Factors of 15: 1, 3, 5, 15
Factors of x2: x*x
Common Factors: 1, 3, 6, x*x
Greatest Common Factor: 6x2
Factoring out a GCF
This is distribution in reverse!
Distribute: 2(x + 3) = 2(x) + 2(3) = 2x + 6
Factoring: 2x + 6 = 2(x) + 2(3) = 2(x + 3)
Factoring out the GCF
• Factor: 3x – 12
• Factor 12y2 – 5y
Factoring out the GCF
Factor: 4x2y3 + 10x5y - 6x3y2
Practice with GCFs
1. 2a + 2b + 2c
2. 3xy – 4xy2 + 5x2y3
3. 7x2 + 21x + 14
A few more…
x
+3
x(
) + 3(
=
)=
x(x – 2) + 3(x – 2) =
Factor by Grouping
This is the same as doing the GCF, but now we do it
THREE times!
• Factor x(z + w) + 2(z + w)
• Example: Factor ad + 3a – d2 – 3d
• Solution:
ad + 3a – d2 – 3d = ad – d2 + 3a – 3d =
= d(a – d) + 3(a – d) =
= (a – d)(d + 3)
Practice with Factoring by Grouping
1. 4x3 + 2x2 – 6x – 3
2. 3xy + 21x - 2y – 14
Factoring Trinomials
• Factoring by Grouping
STEPS:
1. Multiply the first and last term.
2. Identify the coefficient of the x term.
3. Find two numbers that MULTIPLY to the product of the
first and last term (Step 1) and ADD to the coefficient of
the x term (Step 2).
4. Rewrite the original problem, replacing the middle term
with the numbers from Step 3.
5. Factor by grouping.
Example
Factor:
10a2 – a – 2
Steps
We need to find two numbers that:
1) multiply to -20; and
2) add to -1
The only two numbers that work are -5 and
+4.
=10a2 – 5a + 4a – 2
Rewrite the equation by substituting the
new values in for the middle term.
=5a(2a – 1) + 2(2a – 1)
Factor by grouping.
=(2a – 1)(5a + 2)
Factor: x2 – 12x + 32
Factor: 3x2 – 6x + 3
Factor: 4x2 – 14x – 30