Download Lecture notes for Section 5.5

Int. Alg. Notes
Section 5.5
Page 1 of 4
Section 5.5: Factoring Trinomials
Big Idea: Polynomials are the most important topic in algebra because any equation that can be written using
addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be
solved by converting the equation into a polynomial equation.
Looking at the pattern of the quadratic trinomial answers we get when we multiply linear binomials
gives us insight of how to start with a quadratic trinomial and factor it (or “undo” the multiplication) into the
product of two linear binomials.
Big Skill: You should be able to factor nonprime quadratic trinomials into a product of two linear binomials
Let’s look at the patterns we get when we multiply pairs of linear binomials:
 x  2  x  3  x 2  3x  2 x  6
 2 x  3 4 x  5  8x 2  10 x  12 x  15
 x2  5x  6
Notice the following observations:

The answer is a quadratic (degree = 2)
trinomial.

The leading term of the trinomial has a
coefficient of 1, and the coefficients of the x terms in
each binomial also have a coefficient of 1.

The constant term in the trinomial is the
product of the constant terms from each binomial.
(i.e., 23 = 6)

The coefficient of the linear term in the
trinomial is the sum of the constant terms from each
binomial. (i.e., 2 + 3 = 5)
 8 x 2  22 x  15
 x  3 x  4 
Notice the following observations:

The answer is a quadratic (degree = 2)
trinomial.

The coefficient of the leading term of the
trinomial is the product of the coefficients of the x
terms in each binomial. (i.e., 24 = 8)

The constant term in the trinomial is the
product of the constant terms from each binomial.
(i.e., 35 = 15)

The product of the leading and constant terms
of the trinomial is equal to the product of the
coefficients of the linear terms you get from FOIL
(i.e., 815 = 120 and 1012 = 120)
 x  4 2x  7 
Notice the following observations:

Notice the following observations:







 x  5 x  6 
 4x  33x  5 
Use these patterns to factor
Use these patterns to factor
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
x 2  9 x  18 
Section 5.5
Page 2 of 4
6 x 2  13x  15 
Steps for Factoring Quadratic Trinomials of the Form x2 + bx + c:

Find factors of c that sum to b:
o List all factors of b
o Add up each pair of factors; choose the pair of factors that add up to b

Supposing that the factors are m and n (i.e., mn = c), write the trinomial in factored form as:
x2 + bx + c = (x + m)(x + n)

Check your work by multiplying out your factors.
Example:

Factor x2 – 4x – 12

All the factors of -12 are: (+1)(-12), (-1)(+12), (+2)(-6), (-2)(+6), (+3)(-4), (-3)(+4)

The factors that add up to –4 are: (+2) and (-6)

 x2 – 4x – 12 = (x+ 2)(x – 6)
Practice:
1. Factor x2 + 8x + 12
2. Factor a2 – 11a + 18
3. Factor p2 – 2p – 35
4. Factor z2 + 5z + 10
5. Factor 2k3 + 6k2 – 56k
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.5
Page 3 of 4
Steps for Factoring Quadratic Trinomials of the Form ax2 + bx + c:

Find factors of ac that sum to b…
o List all factors of ac
o Add up each pair of factors; choose the pair of factors that add up to b

Supposing that the factors are m and n (i.e., mn = ac and m + n = b), re-write the trinomial with the linear
term broken up into a sum using m and n:
ax2 + bx + c = ax2 + mx + nx+ c

Take your re-written polynomial and factor it by grouping.

Check your work by multiplying out your factors.
Example:

Factor 2x2 – 9x – 18

Multiply leading term coefficient and constant term: (2)(-18) = -36

All the factors of -36 are: (+1)(-36), (-1)(+36), (+2)(-18), (-2)(+18), (+3)(-12), (-3)(+12), (+4)(-9),
(-4)(+9), (+6)(-6)

The factors that add up to -9 are: (+3) and (-12)

2x2 – 9x – 18 = 2x2 + 3x – 12x – 18
= x(2x + 3) – 6(2x + 3) = (x – 6) (2x + 3)
Practice:
6. Factor 3x2 + 11x + 10
7. Factor 6a2 – 5a – 4
8. Factor 15p2 + 26p + 8
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.5
Page 4 of 4
9. Factor 18x2 + 5xy – 2y2
10. Factor -15k3 + 23k2 – 4k
11. Factor 2n4 – 7n2 – 15
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.