Download Ch.5, Section 5

Chapter 5, Section 5
Factoring
The Greatest Common Factor
Factoring by grouping
TEST Three
Difference of Two Squares/Cubes – 5.6
Factoring Trinomials – 5.7
Solving Equations by factoring – 5.9
TEST Four
Now, the reverse process to multiplication: Factoring.
Just like with numbers:
multiply:
2(3)(5)
factor:
12
factor:
6x 3 y 2
multiply/factor
We’ll do this in steps.
First step is always factor out the Greatest Common Factor if there is one, then you apply
steps that we’ll show you in the upcoming sections.
Finding the Greatest Common Factor:
Take each term:
factor it to primes
identify factors that are common to each
multiply these common factors to form the GCF
factor it out of the polynomial to result in a multiplied form
Example
6x 3 y 2  9x 2 y 2  3x 2 y  6x 3 y 2  (1)9x 2 y 2  3x 2 y
I like to use stacks or tables to organize the prime factors initially so I’ll put in a table:
6x 3 y 2
2
3
x
x
x
y
(1)9x 2 y 2
3
x
x
y
3x 2 y
3
x
x
y
GCF
3
x
x
y
3
1
y
It’s not always possible to know how many columns you need but you will always know
how many rows you need: one for each term + one for the GCF. The columns that have
an entry in EVERY row are the ones that contribute a factor to the GCF.
So the GCF is 3x 2 y .
Let’s do another:
2a 3 y 2  8a 2 y 3  4a 2 y 3  2a 2 y 4
 2a 3 y 2  (1)8a 2 y 3  4a 2 y 3  (1)2a 2 y 4
2a 3 y 2
(1)8a 2 y 3
4a 2 y 3
(1)2a 2 y 4
GCF
So the GCF is _______________________________________________.
Sometimes it is not necessary to go to the trouble to make a chart. Look at
4xy  8x
we can write it out factored and see the GCF.
The GCF is ______________________.
Once the GCF is factored out you will use the techniques from 5.6 and 5.7 to finish the
factoring if need be.
Looking at another type of polynomial, let’s explore more factoring.
If you have four terms, you can try factoring by grouping to get it from added form to
multiplied form:
Look at ( a + x) ( b + y). When you multiply these using FOIL, you get
ab + ay + xb + xy.
The O term and the I term are not “like” so they don’t combine at all. In order to factor
this expression, you use a clever technique called “grouping”.
There’s no GCF for the whole expression but we can group the terms and find a
“semi – GCF”:
( ab + ay ) + (xb + xy)
associate pairs of terms that have a factor in
common
a ( b + y) + x ( b + y)
factor out the common terms for each
associated pair
notice that ( b + y) is now a common term and factor it out, getting
(a + x) ( b + y)
Let’s do another following the steps above.
Looking at xy  xq  py + pq, notice that there are 4 terms – the O and the I terms don’t
combine. There’s no obvious common factor but if we’re careful we can group terms and
find a common binomial factor.
(xy  xq) + (py + pq)
note how careful I am with that “minus sign”…I made it go
with the number and put a home made + sign inbetween my
grouped terms
x (y  q) + p ( y + q)
this doesn’t work out. There’s no common binomial factor.
rewrite the problem as
(xy  xq) + (py   pq)
now you can see that if I factor out a p from the second
group I’ll get a common binomial factor of ( x q)
x (y  q) + p (y  q)
( y  q) ( x  p)
done
Let’s try this one:
xy + xm  3y  3m
And this one:
PQ  PT  5Q + 5T