8-3 Factoring x2 + bx + c
Objective
1.Factor quadratic trinomials of the
form x2 + bx + c.
2. Factor quadratic trinomials of the
form x2 + bx + c.
3. Factor four terms by grouping
(front 2, back 2)
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8-3 Factoring x2 + bx + c
Example 1A: Factoring by Using the GCF
Factor each polynomial. Check your answer.
–14x – 12x2
Check –2x(7 + 6x)
–14x – 12x2 
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Multiply to check your answer.
The product is the original
polynomial.
8-3 Factoring x2 + bx + c
Example 1B
Factor each polynomial. Check your answer.
8x4 + 4x3 – 2x2
8x4 = 2  2  2  x  x  x  x
4x3 = 2  2  x  x  x
Find the GCF.
2x2 = 2 
xx
2
x  x = 2x2 The GCF of 8x4, 4x3 and –2x2 is 2x2.
4x2(2x2) + 2x(2x2) –1(2x2) Write terms as products using the
2x2(4x2 + 2x – 1)
Check 2x2(4x2 + 2x – 1)
8x4 + 4x3 – 2x2
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GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original polynomial.
8-3 Factoring x2 + bx + c
Example 2
Factor each expression.
a. 4s(s + 6) – 5(s + 6)
4s(s + 6) – 5(s + 6)
(4s – 5)(s + 6)
The terms have a common
binomial factor of (s + 6).
Factor out (s + 6).
b. 7x(2x + 3) + (2x + 3)
7x(2x + 3) + (2x + 3)
The terms have a common
binomial factor of (2x + 3).
7x(2x + 3) + 1(2x + 3) (2x + 1) = 1(2x + 1)
(2x + 3)(7x + 1)
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Factor out (2x + 3).
8-3 Factoring x2 + bx + c
You may be able to factor a polynomial by
grouping. When a polynomial has four terms,
you can make two groups and factor out the
GCF from each group.
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8-3 Factoring x2 + bx + c
Example 3A: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
6h4 – 4h3 + 12h – 8
(6h4 – 4h3) + (12h – 8) Group terms that have a common
number or variable as a factor.
2h3(3h – 2) + 4(3h – 2) Factor out the GCF of each
group.
2h3(3h – 2) + 4(3h – 2) (3h – 2) is another common
factor.
(3h – 2)(2h3 + 4)
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Factor out (3h – 2).
8-3 Factoring x2 + bx + c
Example 3A Continued
Factor each polynomial by grouping.
Check your answer.
Check (3h – 2)(2h3 + 4)
Multiply to check your
solution.
3h(2h3) + 3h(4) – 2(2h3) – 2(4)
6h4 + 12h – 4h3 – 8
6h4 – 4h3 + 12h – 8
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The product is the original
polynomial.
8-3 Factoring x2 + bx + c
Example 43B: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
(5y4 – 15y3) + (y2 – 3y)
Group terms.
5y3(y – 3) + y(y – 3)
Factor out the GCF of
each group.
5y3(y – 3) + y(y – 3)
(y – 3) is a common factor.
(y – 3)(5y3 + y)
Factor out (y – 3).
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8-3 Factoring x2 + bx + c
Example 4: Factoring with Opposites
Factor 2x3 – 12x2 + 18 – 3x
2x3 – 12x2 + 18 – 3x
(2x3 – 12x2) + (18 – 3x)
2x2(x – 6) + 3(6 – x)
2x2(x – 6) + 3(–1)(x – 6)
2x2(x – 6) – 3(x – 6)
(x – 6)(2x2 – 3)
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Group terms.
Factor out the GCF of
each group.
Write (6 – x) as –1(x – 6).
Simplify. (x – 6) is a
common factor.
Factor out (x – 6).
8-3 Factoring x2 + bx + c
Lesson Quiz: Part 1
Factor each polynomial. Check your answer.
1. 16x + 20x3
4x(4 + 5x2)
2. 4m4 – 12m2 + 8m
4m(m3 – 3m + 2)
Factor each expression (by grouping).
3. 3y(2y + 3) – 5(2y + 3)
(2y + 3)(3y – 5)
4. 2x3 + x2 – 6x – 3
(2x + 1)(x2 – 3)
5. 7p4 – 2p3 + 63p – 18
(7p – 2)(p3 + 9)
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8-3 Factoring x2 + bx + c
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) = x2 + 7x + 10
Remember!
When you multiply two binomials, multiply:
First terms
Outer terms
Inner terms
Last terms
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8-3 Factoring x2 + bx + c
The guess and check method is usually not the
most efficient method of factoring a trinomial. Look
at the product of (x + 3) and (x + 4).
x2
12
(x + 3)(x +4) = x2 + 7x + 12
3x
4x
The coefficient of the middle term is the sum of 3
and 4. The third term is the product of 3 and 4.
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8-3 Factoring x2 + bx + c
Holt Algebra 1
8-3 Factoring x2 + bx + c
Example 1: Factoring x2 + bx + c
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 + x + 5x + 5
= x2 + 6x + 5
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Use the FOIL method.
The product is the
original polynomial.
8-3 Factoring x2 + bx + c
Example 2: Factoring x2 + bx + c
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 – 3x – 5x + 15 Use the FOIL method.
= x2 – 8x + 15 
The product is the
original polynomial.
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8-3 Factoring x2 + bx + c
Example 3: Factoring x2 + bx + c
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)
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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.
8-3 Factoring x2 + bx + c
Example 4
Factor each trinomial. Check your answer.
x2 + 2x – 15
(x +
)(x +
)
Factors of –15 Sum
–1 and 15
14 
–3 and 5
2
(x – 3)(x + 5)
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b = 2 and c = –15; look for
factors of –15 whose sum is 2.
The factor with the greater
absolute value is positive.
The factors needed are –3 and
5.
8-3 Factoring x2 + bx + c
Lesson Quiz: Part I
Factor each trinomial.
1. x2 – 11x + 30 (x – 5)(x – 6)
2. x2 + 10x + 9
(x + 1)(x + 9)
3. x2 – 6x – 27
(x – 9)(x + 3)
4. x2 + 14x – 32
(x + 16)(x – 2)
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