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Factoring polynomials will be the
main topic on Exam 2; we will be
factoring polynomials in Lessons 6
and 7, and factoring will come up
again quite a bit in Lesson 8,
somewhat in Lesson 9, and once or
twice in Lesson 10. You will still
need to understand the other
topics that we will cover (such as
solving linear equations in Lesson
10), but factoring polynomials will
make-up about 75% of Exam 2, so
please be prepared.
Throughout this course we will be looking at how to undo different
operations in algebra. When covering exponents we showed how
3
= − 27 , then when covering radicals we saw how to get back to the
original base of − 3 by undoing an exponent of 3 with a cubed root
3
= − 3 ). When covering polynomial multiplication we showed
how to multiply factors such as
and
to obtain a product
which is a polynomial
2
, and in these notes we will show
how to undo that product to get back to the original factors.
Factoring Polynomials:
- finding a product that is equivalent to some original polynomial
o 4 𝑥
2
− 17 𝑥 − 15 is equivalent to
- we will be using factoring as a way of undoing polynomial
multiplication (going from a sum of terms to product of factors)
o in the trinomial 4 𝑥
2
2
, − 17 𝑥, and − 15 are all
terms, while in the product
and
are both factors
- not all polynomials are factorable
o 𝑥
2
- 2 𝑥 + 3 is an example of a polynomial that is prime, which
means that trinomial cannot be expressed in factored form
Greatest Common Factor (GCF):
- the largest factor that is common to each term of an expression
- the GCF of an expression could be a number, a variable, a quantity,
or some combine of the three
o in the binomial 6 𝑥
2
4
3
2
the GCF is 3 𝑥
2
2
o when the GCF is factored out, each term in the polynomial is
divided by the 3 𝑥
2
2
2
4
3
2
2
2
6 𝑥
2
𝑦
4
3 𝑥
2
𝑦
2
9 𝑥
3
𝑦
2
3 𝑥
2
𝑦
2
𝟐
𝟐
𝟐
Step one is identifying the GCF, and step two is dividing it out.
The GCF of an expression does not have to be simply a number or a
variable. As stated before, the GCF could be a quantity as well, as we’ll
see in the next example:
2
3
the GCF
is
- when the GCF is factored out, each term in the polynomial is divided
by the
2
3
2
3
2
Once the GCF of (𝑥 − 1 ) is factored out, we cannot factor the remaining
polynomial any further. So we go back to what we did in the previous
lesson; multiply the polynomials and combine like terms.
2
2
2
2
2
𝟐
Keep in mind that when a GCF is factored out, we don’t list it more than
once. For instance when
is factored out of the binomial
, we have
, NOT
. The same is true if we factor a 𝑦 from the
binomial 3 𝑥𝑦 − 4 𝑦 to get 𝑦( 3 𝑥 − 4 ); both terms had a common factor of
𝑦, but when we factor it out we only have one factor of 𝑦 as the GCF.
After distributing
factors and
combining like
terms, the
resulting
polynomial can
sometimes be
factored further.
That is not the
case on this
problem, but it
could be with
other problems.
Example 2 : Factor the following polynomials by taking out the GCF, and
write each final answer in factored form.
a.
(𝑥+ 1 )(𝑥− 2 )
( 𝑥+ 1
)
(𝑥+ 1 )(𝑥+ 3 )
( 𝑥+ 1
)
b.
(𝑥+𝑦)(𝑥− 9 )
( 𝑥+𝑦
)
( 9 𝑥− 1 )(𝑦+𝑥)
( 𝑥+𝑦
)
c. 2
3
2
2
2
( 1 −𝑥
)
3
( 1 −𝑥
)
2
3 𝑥
( 1 −𝑥
)
2
( 1 −𝑥
)
2
2
2
𝟐
Keep in mind that the
binomial
must
have parentheses around it
so that the negative sign in
front of it gets distributed
to both terms ( 9 𝑥 and − 1 ).
Had there been a plus sign
in front of the binomial
9 𝑥 − 1 , parentheses would
not have been required, but
they could still be included
just for consistency.
