Extracting Factors from Polynomials

1
Extracting Factors from Polynomials

• Learn to extract the greatest common factor from
  a polynomial.

2
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.

3
Finding the GCF

• To find the greatest common factor (GCF) of two
  (or more) terms in a polynomial
• Find the prime factorization of the coefficient
  of each term and then expand each monomial term.
• Find all of the common factors.
• Multiply these common factors together to get the
  greatest common factor (GCF).

4
Prime Factorization

• To review how to find the prime factorization of
  a number, lets look at a couple of examples.
• 1. 2.

Prime Factorization of 75 is 355
Prime Factorization of 45 is 335
5
Expanding a Monomial

• To expand a monomial, we find the prime
  factorization of the coefficient, and write the
  variables without exponents.
• For example
• 24x2y3
• 15a2b
• 8xyz

2 2 2 3 x x y y y
3 5 a a b
2 2 2 x y z
6
Finding the GCF

• To find the GCF of the terms in the polynomial,
  expand each term and find the common factors
• Lets look at this example

15x 45x2
15x 3 5 x
45x2 3 3 5 x x
GCF 3 5 x 15x
7
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.

45x2 3 3 5 x x
15x 3 5 x
The GCF was 15x
1
3x
15x ( )
15x 45x2
8
Finding the GCF

• Lets 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
2n 2
3n
4
4n 4 6n 3 8n 2
2n 2( )