Download Section 4.3 FACTORING Factoring is the

Section 4.3
FACTORING
Factoring is the opposite of multiplication.
Recall using a factor tree to find prime factors.
Completely factor the following number by writing it as the product of only prime factors.
54 =
Similarly, we want to rewrite a polynomial as the product of prime polynomial factors. When a polynomial is rewritten
with only prime factors, we call the polynomial completely factored.
Greatest Common Factor or “GCF.”
We start the factoring process of polynomials by factoring out the Greatest Common Factor or “GCF.”
The greatest common factor of a group of numbers is the largest factor that can divide evenly into all of those numbers.
For example,
the GCF of 20, 8, and, 12 is ____________.
The GCF of 18, 24, and 36 is ______________.
The GCF of 9, 27, and 36 is _________________.
Factoring Polynomials: Step 1- The Greatest Common Factor
For a polynomial, factoring out the GCF means to do the distribution process in reverse. We will rewrite a polynomial as
GCF(what is left over from each term)
Example 1: Factor out the GCF of the following.
a) 18𝑥 3 − 24𝑥 2
When looking at the variables how many x’s are common to each term?
Pull out a common variable
1) only if it is in each term and
2) use the smallest exponent that we see on that variable.
b) 9𝑥 3 − 27𝑥 2 + 36𝑥
c) 20𝑥 2 𝑦 4 − 8𝑥 3 𝑦 + 12𝑥 4 𝑦 2
d) 15𝑥 4 𝑦 5 − 20𝑥 3 𝑦 3 + 5𝑥 2
e) 7𝑥 2 + 10𝑦 3 𝑧
We can also factor out a common binomial factor that appears in each term.
Example 2: Factor out the GCF of the following.
a)
3𝑎(𝑥 + 7𝑦) + 2𝑏(𝑥 + 7𝑦)
b)
2𝑥 2 (6𝑦 + 5) + 7𝑥(6𝑦 + 5) + 8(6𝑦 + 5)
c) 45𝑥 4 𝑦 3 (𝑦 − 2𝑥) − 30𝑥𝑦 2 (𝑦 − 2𝑥)
Step 2: Count the terms leftover after removing the GCF.
If there are four terms:
Use grouping method to factor.
How to Factor 4 terms by Grouping
To factor by grouping:
1) Group together the first two terms and the last two terms.
2) Factor out GCF of each group separately.
3) Factor out common binomial.
*If there is no common binomial, then rearrange the terms and start over.
Example 3: Completely factor the following polynomials.
a) 𝑥 2 𝑦 + 9𝑥 2 + 5𝑦𝑧 2 + 45𝑧 2
c) 16𝑥 2 𝑦 − 40𝑥 2 − 10 + 4𝑦
b) 5𝑎𝑦 − 5𝑏𝑦 − 3𝑎 + 3𝑏