103: Factoring Trinomials

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10-3 Factoring Trinomials
OBJECTIVE You must factor quadratic trinomials.
There have been many ways created to factor
trinomials over the years. The method presented
in these slides will work for any trinomial that
can be factored. If the method does not work,
the trinomial can not be factored. You need to
learn these steps. We start with an example.
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10-3 Factoring Trinomials
EXAMPLE 1 Factor 10x2 - 27x 18.
1 x 180
-1 x -180
2 x 90
-2 x -90
1) Take the coefficient off squared-term and
multiply it by the constant term.
3 x 60
-3 x -60
4 x 45
-4 x -45
2) List all the factor pairs of 180. You need
to include both positive and negative factors.
x
180

5 x 36
-5 x -36
6 x 30
-6 x -30
-12 -15 -27
3) Find the factor-pair that adds to the
coefficient of the middle term. This is your
magic pair.
9 x 20
-9 x -20
10 x 18
-10 x -18
10x2 - 27x 18
12 x 15
-12 x -15
4) Replace the middle term by putting the
variable on both numbers in the magic pair.
10x2 - 12x - 15x 18
2x(5x - 6)
3(-5x 6)

5) Now, you have a 4-nomial. Factor it using
the skills from the last section.
2x(5x - 6) - 3(5x - 6) (5x - 6)(2x - 3)
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10-3 Factoring Trinomials
This process always works, although you may need
to re-arrange terms. You should also pull out any
GCFs before beginning trinomial factoring.
EXAMPLE 2 Factor 14t - 36 2t2.
2t2 14t - 36 2(t2 7t - 18)
Re-arrange the terms in descending order and pull
out GCF.
Factor the trinomial.
t2 7t - 18
1) Take the coefficient off squared-term and
multiply it by the constant term (keeping signs
of each term.)
1 x -18
-1 x 18
x

-18
2 x -9
-2 x 9
3 x -6
-3 x 6
2) List all the factor pairs of -18. You need
to include both positive and negative factors.
-2 9 7
3) Find the factor-pair that adds to the
coefficient of the middle term. This is your
magic pair.
t2 7t - 18
t2 - 2t 9t - 18
4) Replace the middle term by putting the
variable on both numbers in the magic pair.
t(t - 2)
9(t - 2)

2(t - 2)(t 9)
(t - 2)(t 9)
5) Now, you have a 4-nomial. Factor it using
the skills from the last section.
You have to put the GCF back on your answer.
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10-3 Factoring Trinomials
If there is no magic pair for the trinomial, then
the polynomial is prime. You can do nothing with
prime polynomials for now.
EXAMPLE 3 Factor 2a2 - 11a 7.
2a2 - 11a 7
1) Take the coefficient off squared-term and
multiply it by the constant term (keeping signs
of each term.)
1 14 15
-1 -14 -15
1 x 14
-1 x -14
x

14
2) List all the factor pairs of 14. You need to
include both positive and negative factors.
2
7
2 x 7
-2 x -7
3) Find the factor-pair that adds to the
coefficient of the middle term. This is your
magic pair.
-2 -7 -9
2 7 9
Notice there is no magic pair. Therefore the
trinomial is a prime trinomial.
Answer prime polynomial
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10-3 Factoring Trinomials
HOMEWORK
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