Transcript Prime Factorization and GCF

Extracting Factors from
Polynomials
Learn to extract the
greatest common factor
from a polynomial.
Extracting Factors
• To factor a polynomial, we first begin by
determining if the polynomial has a monomial
factor other than 1.
• We need to check to see if the terms of the
polynomial have a GCF (greatest common factor).
• If so, we can extract that monomial factor by
dividing the polynomial by that factor.
• The quotient from that division is the second
factor of the polynomial.
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Finding the GCF
To find the greatest common factor (GCF) of two
(or more) terms in a polynomial:
1. Find the prime factorization of the
coefficient of each term and then expand
each monomial term.
2. Find all of the common factors.
3. Multiply these common factors together to
get the greatest common factor (GCF).
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Prime Factorization
To review how to find the prime factorization of
a number, let’s look at a couple of examples.
1. 45
5
2. 75
3
3
Prime Factorization
of 45 is 3·3·5
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25
3
9
5
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Prime Factorization
of 75 is 3·5·5
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Expanding a Monomial
• To expand a monomial, we find the prime
factorization of the coefficient, and
write the variables without exponents.
• For example:
24x2y3 =
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2·2·2·3·x·x·y·y·y
15a2b =
3·5·a·a·b
8xyz =
2·2·2·x·y·z
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Finding the GCF
• To find the GCF of the terms in the polynomial,
expand each term and find the common factors:
• Let’s look at this example:
15x + 45x2
15x = 3 · 5 · x
45x2 = 3 · 3 · 5 · x · x
GCF = 3 · 5 · x = 15x
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Factoring a Polynomial
• Once you have found the GCF, that will be the
first factor. It is written in front of a set of
parentheses for the paired factor.
• The numbers and variables that are left after the
GCF has been removed go on the inside of the
parentheses. This becomes the paired factor.
15x = 3 · 5 · x
45x2 = 3 · 3 · 5 · x · x
The GCF was 15x
15x + 45x2 = 15x ( 1 +3x )
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Finding the GCF
• Let’s try another example
4n 4 + 6n 3 – 8n 2
6n 3 = 2 · 3 · n · n · n
4n 4 = 2 · 2 · n · n · n · n
8n 2 = 2 · 2 · 2 · n · n
GCF = 2 · n · n = 2n 2
4n 4 + 6n 3 – 8n 2 = 2n 2( 2n 2 + 3n – 4)
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