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Factoring x2 + bx + c
Section 8-5
Goals
Goal
• To factor trinomials of the
form x2 + bx + c.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• None
Factoring a Trinomial
x2 + bx + c
Earlier you learned how to multiply two binomials using the
Distributive Property or the FOIL method. In this lesson, you
will learn how to factor a trinomial into two binominals.
Notice that when you multiply (x + 2)(x + 5), the constant term
in the trinomial is the product of the constants in the binomials.
(x + 2)(x + 5) =
2
x
+ 7x + 10
Factoring a Trinomial
2
x + bx + c
To factor a simple trinomial of the form
x 2 + bx + c (leading coefficient is 1),
express the trinomial as the product of
two binomials. For example,
x 2 + 10x + 24 = (x + 4)(x + 6).
Factoring these trinomials is based on
reversing the FOIL process.
Factoring a Trinomial
x2 + bx + c
Look at the product of (x + a) and (x + b).
x2
ab
(x + a)(x + b) = x2 + ax + bx + ab
ax
bx
= x2 + (a + b)x + ab
The coefficient of the middle term is the sum of a and b. The
constant term is the product of a and b.
Factoring a Trinomial
2
x + bx + c
Recall by using the FOIL method that
F
O
I
L
(x + 2)(x + 4) = x 2 + 4x + 2x + 8
= x 2 + 6x + 8
To factor x 2 + bx + c into (x + one #)(x + another #), note that
b is the sum of the two numbers and c is the product of the
two numbers.
So we’ll be looking for 2 numbers whose product is c and
whose sum is b.
Note: there are fewer choices for the product, so that’s why
we start there first.
Procedure for Factoring Trinomials
of the Form x 2 + bx + c
Factor: x 2  12 x  20
1. List the factors of 20 (two numbers that 20
1×20
multiply to equal the constant):
2×10
2. Select the pairs those sum is 12 (the
4×5
middle term).
3. Write the two
 x  10  x  2 
binomial factors:
2
4. Check using FOIL: x  2 x  10 x  20
x 2  12 x  20
Factoring a Trinomial
2
x + bx + c
Factoring Trinomials x2 + bx + c
TIP
When c is positive, its factors have the same
sign. The sign of b tells you whether the
factors are positive or negative. When b is
positive, the factors are positive and when b
is negative, the factors are negative.
x  12 x  20
x  9 x  20
 x  10 x  2
 x  5 x  4
2
2
Example: c is Positive
Factor each trinomial. Check your answer.
x2 + 6x + 5
(x + )(x + )
b = 6 and c = 5; look for factors of 5
whose sum is 6.
Factors of 5 Sum
1 and 5
6
The factors needed are 1 and 5.
(x + 1)(x + 5)
Check (x + 1)(x + 5) = x2 + 5x + x + 5
= x2 + 6x + 5
Use the FOIL method.
The product is the
original trinomial.
Example: c is Positive
Factor each trinomial. Check your answer.
x2 + 6x + 9
(x + )(x + ) b = 6 and c = 9; look for factors of 9
whose sum is 6.
Factors of 9 Sum
1 and 9
10 
3 and 3
6  The factors needed are 3 and 3.
(x + 3)(x + 3)
Check (x + 3)(x + 3 ) = x2 + 3x + 3x + 9 Use the FOIL method.
The product is the
2

= x + 6x + 9
original trinomial.
Example: c is Positive
Factor each trinomial. Check your answer.
x2 – 8x + 15
(x + )(x + )
b = –8 and c = 15; look for
factors of 15 whose sum is –8.
Factors of –15 Sum
–1 and –15 –16 
–3 and –5 –8  The factors needed are –3 and –5 .
(x – 3)(x – 5)
Check (x – 3)(x – 5 ) = x2 – 5x – 3x + 15 Use the FOIL method.
The product is the
2
= x – 8x + 15 
original trinomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 + 8x + 12
(x + )(x + )
Factors of 12 Sum
1 and 12 13
2 and 6
8
b = 8 and c = 12; look for factors of
12 whose sum is 8.
The factors needed are 2 and 6 .
(x + 2)(x + 6)
Check (x + 2)(x + 6 ) = x2 + 6x + 2x + 12 Use the FOIL method.
= x2 + 8x + 12  The product is the
original trinomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 – 5x + 6
(x + )(x+ )
b = –5 and c = 6; look for
factors of 6 whose sum is –5.
Factors of 6 Sum
–1 and –6 –7
–2 and –3 –5  The factors needed are –2 and –3.
(x – 2)(x – 3)
Check (x – 2)(x – 3) = x2 – 3x – 2x + 6 Use the FOIL method.
= x2 – 5x + 6 
The product is the
original trinomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 + 13x + 42
(x + )(x + )
Factors of 42
1 and 42
2 and 21
6 and 7
b = 13 and c = 42; look for
factors of 42 whose sum is
13.
Sum
43 
23 
13  The factors needed are 6 and 7.
(x + 6)(x + 7)
Check (x + 6)(x + 7) = x2 + 7x + 6x + 42 Use the FOIL method.
= x2 + 13x + 42  The product is the
original trinomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 – 13x + 40
b = –13 and c = 40; look for factors
of 40 whose sum is –13.
(x + )(x+ )
Factors of 40 Sum
–2 and –20 –22 The factors needed are –5 and –8.
–4 and –10 –14
–5 and –8
–13
(x – 5)(x – 8)
Check (x – 5)(x – 8) = x2 – 8x – 5x + 40
= x2 – 13x + 40
Use the FOIL method.
The product is the
original polynomial.
Factoring Trinomials x2 + bx + c
TIP
When c is negative, its factors have opposite
signs. The sign of b tells you which factor is
positive and which is negative. The factor
with the greater absolute value has the same
sign as b.
x  9 x  22
2
 x  11 x  2
Example: c is Negative
Factor each trinomial.
x2 + x – 20
(x + )(x + )
Factors of –20 Sum
–1 and 20
19 
–2 and 10
8
–4 and 5
1
(x – 4)(x + 5)
b = 1 and c = –20; look for
factors of –20 whose sum is
1. The factor with the greater
absolute value is positive.
The factors needed are 5 and
–4.
Example: c is Negative
Factor each trinomial.
x2 – 3x – 18
(x + )(x + )
Factors of –18 Sum
1 and –18 –17
2 and – 9 – 7 
3 and – 6 – 3 
(x – 6)(x + 3)
b = –3 and c = –18; look for
factors of –18 whose sum is
–3. The factor with the
greater absolute value is
negative.
The factors needed are 3 and
–6.
Helpful Hint
If you have trouble remembering the rules for which
factor is positive and which is negative, you can try
all the factor pairs and check their sums.
Your Turn:
Factor each trinomial. Check your answer.
x2 + 2x – 15
(x + )(x + )
b = 2 and c = –15; look for
factors of –15 whose sum is 2.
The factor with the greater
absolute value is positive.
Factors of –15 Sum
–1 and 15
14 
–3 and 5
2  The factors needed are –3 and
5.
(x – 3)(x + 5)
Check (x – 3)(x + 5) = x2 + 5x – 3x – 15 Use the FOIL method.
= x2 + 2x – 15 
The product is the
original polynomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 – 6x + 8
(x + )(x + )
Factors of 8
–1 and –6
–2 and –4
(x – 2)(x – 4)
Sum
–7
–6 
b = –6 and c = 8; look for
factors of 8 whose sum is –6.
The factors needed are –4
and –2.
Check (x – 2)(x – 4) = x2 – 4x – 2x + 8
= x2 – 6x + 8
Use the FOIL method.
The product is the
original polynomial.
Your Turn:
Factor each trinomial. Check your answer.
x2 – 8x – 20
(x + )(x + )
Factors of –20 Sum
1 and –20 –19
2 and –10 –8 
(x – 10)(x + 2)
b = –8 and c = –20; look for
factors of –20 whose sum is
–8. The factor with the
greater absolute value is
negative.
The factors needed are –10
and 2.
Check (x – 10)(x + 2) = x2 + 2x – 10x – 20
= x2 – 8x – 20

Use the FOIL method.
The product is the
original polynomial.
Practice
More Practice
Assignment
• 8-5 Exercises Pg. 518 - 520: #10 – 46 even