Unit 3:Factors & Products
Math 1201
Handout #2
Common Factors of Polynomials (section 3.3)
• Definitions
1. To Factor a Polynomial
• Means to write it as a product of other polynomials.
• Factoring is the reverse of expanding, or multiplying, polynomials
• In this unit we will learn the following types of factoring:
(1) Common Monomial Factor
(2) Trinomials of the form 𝑥 2 + 𝑏𝑥 + 𝑐
(3) Trinomials of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
(4) Difference of Two Perfect Squares 𝑎2 − 𝑏 2
(5) Perfect Square Trinomials 𝑎2 + 2𝑎𝑏 + 𝑏 2
(6) Trinomials with Two Variables
(7) A Combination of Type (1) and Either other Type (2) to (6)
• On page 151 of the textbook multiplying and factoring are compared
2. Common Factor
• A number that divides evenly into a set of given numbers
• Ex. 3 is a common factor of 9, 15, and 21
• In the last unit we found the greatest common factor between numbers
3. Common Monomial Factor
• A single term (monomial) that divides into each term of a polynomial
• Ex. 4𝑦 is a common monomial factor of 8𝑥 2 𝑦 + 4𝑥𝑦 + 12𝑦
• To find the Common Monomial Factor of a Polynomial
• Find the GCF of the coefficients
• Find the GCF of the variables
Find the variables that are common in the terms
Take the lowest exponent on each of these variables
• Divide the common monomial factor out of each term of the polynomial
• Write the polynomial as a product of these factors
• A polynomial is fully factored when it cannot be factored any further
Ex.
4𝑚 + 12 = 2(2𝑚 + 6)
Ex.
4𝑚 + 12 = 4(𝑚 + 3)
The second example shows the fully factored form of the polynomial
4𝑚 + 12 because in the first example there is still a common factor left
in the bracket
• If a polynomial begins with a negative sign, the negative is factored out
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Unit 3:Factors & Products
Math 1201
Handout #2
as part of the common monomial factor and signs are changed in the
remaining factor
• We will look at the Algebraic Method and the "Window" Method
• Examples and Problems
1. State the common monomial factor for each of the following
a) 𝑥 2 and 𝑥 6
b) 𝑡 4 and 𝑡 7
c) 2𝑥 2 and 12𝑥
d) 𝑥 2 and 𝑥 6
e) 𝑢2 𝑣 and 𝑢3 𝑣 2
f) 𝑥 6 𝑦 4 and − 𝑥𝑦
g) 9𝑦 8 𝑧 4 and − 12𝑦 5 𝑧 4
h) −15𝑥 6 𝑦 3 and 45𝑥𝑦 3
i) 14𝑥 2 , 1, and 𝑥 6
j) 5𝑦 4 and 10𝑥 2 𝑦 2
k) 28𝑎4 𝑏 2 , 14𝑎3 , and 42𝑎2 𝑏 5
l) 16𝑥 2 𝑦, 12𝑥𝑦 2 , and 36𝑥 2
2. Factor by removing the greatest common monomial factor
a) 3𝑥 + 3
b)
5𝑦 − 5
c) 6𝑥 + 36
d)
4𝑥 − 28
e) 8𝑡 − 16
f)
4𝑢 − 12
g) 25𝑥 − 10
h)
14𝑦 − 7
i) 24𝑦 2 − 18
j)
8𝑥 3 + 12
k) 𝑥 2 + 𝑥
l)
𝑥3 − 𝑥
m) 25𝑢2 − 14𝑢
n)
36𝑡 4 + 24𝑡 2
o) 2𝑥 4 + 6𝑥 3
p)
9𝑧 6 + 27𝑧 4
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Unit 3:Factors & Products
Math 1201
Handout #2
q) −27𝑥 2 + 9𝑦 2
r)
−12𝑥 2 − 5𝑥 3
s) −12𝑥 2 − 2𝑥
t)
−12𝑢7 + 9𝑢5
u) 12𝑥 2 + 16𝑥 − 8
v)
9 − 3𝑦 − 15𝑦 2
w) 21𝑥 2 𝑦 5 + 35𝑥 6 𝑦 3 − 14𝑥 5 𝑦 4
x)
−16𝑥 5 𝑦 3 + 8𝑥 2 𝑦 − 24𝑥 4 𝑦 5
y) −15𝑚4 𝑛3 − 25𝑚7 𝑛 + 30𝑚6 𝑛8
z)
12𝑥 3 𝑦 2 − 6𝑥 2 𝑦 + 18𝑥 4 𝑦 5
Textbook: Pages 155 - 156 #7, 8, 9, 10, 15(b), 16
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