Download Factoring: The GCF and Factor By Grouping

Factoring: The GCF and Factor By Grouping
Definition • To factor a polynomial is to write the polynomial as a product of two (or more) other polynomials.
• When a polynomial is written as a product, each polynomial
being multiplied in that product is called a factor.
• A prime polynomial is a polynomial that cannot be factored using only integer coefficients.
Remark 1 You’ve been exposed to factoring before. A whole number can be written as a product of its prime factors.
ex.
−→ 21 = 7 · 3
↖↗
factors of 21!
In polynomials, the idea is similar but, unfortunately, more complicated. To factor a polynomial you have to (somehow) think of two
other polynomials that multiply out to the original polynomial!
Remark 2 In general factoring is the reverse of multiplying.
→ Multiplying →
ex.
−→
4x · .(3x + 5) = 12x2 .+ 20x ..
← Factoring ←
That is, if the problem says, “Multiply: 4x · (3x + 5)” then the solution
would be “12x2 + 20x”, which would be achieved by distributing.
On the other hand if the problem stated, “Factor 12x2 + 20x” then the
solution would be “4x · (3x + 5)”, which would be achieved by reverse
distributing.
Remark 3 In this topic we will learn about the first two factoring procedures,
1. factoring out the GCF and 2. factor by grouping. These
two procedures are crucial building blocks and cannot be glossed over.
Spend extra time studying and understanding these procedures.
Definition The greatest common factor (GCF) between a list of
integers is the largest factor that those integers all have in common.
Remark 4 To find the GCF between two numbers, factor each number into its
prime factors. The overlap is the GCF.
ex.
−→ the GCF between 42 and 36 is 6 because:
42 = 2 · 3 · 7
36 = 2 · 2 · 3 · 3
The overlap is 6.
ex.
−→ the GCF between x4 and x7 is x4 becuase:
x7 = x · x · x · x · x · x · x
x4 = x · x · x · x
The overlap is x4 .
Procedure To factor a polynomial by “factoring out the GCF”:
1 Find the GCF between all the terms in the polynomial.
→ Remember, terms are the things connected by +/− signs!
2 Rewrite each term as a product with the GCF as one of the
factors.
3 Pull the GCF outside of a pair of parenthesis by reverse
distributing.
Procedure To factor a 4 - term polynomial by “factoring by grouping”:
1 Group the first two terms and the last two terms by placing parenthesis around those pairs of terms.
→ Sometimes, you may need to reorder the terms first.
2 From each group formed in step 1, factor out the GCF.
→ Most likely, you will get a different GCF for each group.
3 Factor out the “overall” GCF.
→After step 2 there should be a binomial appearing twice in
the polynomial. This is the “overall” GCF.
Main Idea Become accustomed to the above procedures. Know them inside
and out!
Example 1 Find the GCF for the list of terms. 25p5r7, 30p7r8, 50p5r3.
Example 2 For each polynomial, factor out the GCF.
a 6m2 + 15m
b 36p6q + 45p5q 4 + 81p3q 2
c r(x + 5) − t(x + 5)
Example 3 For each polynomial, factor by grouping.
a 8x3 + 4x2 + 10x + 5
b 7y − 9x − 3xy + 21