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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.
© William James Calhoun, 2001
10-3: Factoring Trinomials
EXAMPLE 1: Factor 10x2 - 27x + 18.
1) Take the coefficient off squaredterm and multiply it by the constant
term.
2) List all the factor pairs of 180.
You need to include both positive
and negative factors.
10 x 18 = 180
3) Find the factor-pair that adds to
the coefficient of the middle term.
This is your “magic pair.”
-12 + -15 = -27
4) Replace the middle term by
putting the variable on both numbers
in the magic pair.
5) Now, you have a 4-nomial.
Factor it using the skills from the last
section.
10x2 - 27x + 18
1 x 180
-1 x -180
2 x 90
-2 x -90
3 x 60
-3 x -60
4 x 45
-4 x -45
5 x 36
-5 x -36
6 x 30
-6 x -30
9 x 20
-9 x -20
10 x 18
-10 x -18
12 x 15
-12 x -15
10x2 - 12x - 15x + 18
2x(5x - 6) + 3(-5x + 6)
2x(5x - 6) - 3(5x - 6)
(5x - 6)(2x - 3)
© William James Calhoun, 2001
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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.
Re-arrange the terms in descending order and
= 2t2 + 14t - 36
pull out GCF.
= 2(t2 + 7t - 18)
Factor the trinomial.
1 t2 + 7t - 18
1) Take the coefficient off squared-term and
multiply it by the constant term (keeping signs
of each term.)
2) List all the factor pairs of -18. You need to
include both positive and negative factors.
3) Find the factor-pair that adds to the
coefficient of the middle term. This is your
“magic pair.”
4) Replace the middle term by putting the
variable on both numbers in the magic pair.
5) Now, you have a 4-nomial. Factor it using
the skills from the last section.
1 x -18 = -18
-2 + 9 = 7
1 x -18
-1 x 18
2 x -9
-2 x 9
3 x -6
-3 x 6
t2 + 7t - 18
t2 - 2t + 9t - 18
t(t - 2) + 9(t - 2)
(t - 2)(t + 9)
2(t - 2)(t + 9)
You have to put the GCF back on your answer.
© William James Calhoun, 2001
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.
1) Take the coefficient off squared-term and
= 2a2 - 11a + 7
multiply it by the constant term (keeping signs
1 + 14 = 15
-1 + -14 = -15
of each term.)
2) List all the factor pairs of 14. You need to
include both positive and negative factors.
2 x 7 = 14
3) Find the factor-pair that adds to the
coefficient of the middle term. This is your
“magic pair.”
1 x 14
-1 x -14
2x7
-2 x -7
2+7=9
-2 + -7 = -9
Notice there is no magic pair.
Therefore the trinomial is a prime trinomial.
Answer: prime polynomial
© William James Calhoun, 2001
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10-3: Factoring Trinomials
HOMEWORK
Page 579
#23 - 41 odd
© William James Calhoun, 2001
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