6.2 Factoring Trinomials
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What You Will Learn
 Factor trinomials of the form x2 + bx + c
 Factoring trinomials in two variables
 Factor trinomials completely
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Factoring Trinomials of the Form
x2 + bx + c
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Factoring Trinomials of the Form x2 + bx + c
You know that the product of two binomials is often a
trinomial. Here are some examples.
Factored Form
F
O
I
L
Trinomial Form
(x – 1)(x + 5) = x2 + 5x – x – 5 = x2 + 4x – 5
(x – 3)(x – 3) = x2 – 3x – 3x + 9 = x2 – 6x + 9
(x + 5)(x + 1) = x2 + x + 5x + 5 = x2 + 6x + 5
(x – 2)(x – 4) = x2 – 4x – 2x + 8 = x2 – 6x + 8
Try covering the factored forms in the left-hand column
above.
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Factoring Trinomials of the Form x2 + bx + c
Can you determine the factored forms from the trinomial
forms? In this section, you will learn how to factor trinomials
of the form x2 + bx + c.
To begin, consider the following factorization.
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Factoring Trinomials of the Form x2 + bx + c
So, to factor a trinomial x2 + bx + c into a product of two
binomials, you must find two numbers m and n whose
product is c and whose sum is b.
There are many different techniques that can be used to
factor trinomials.
The most common technique is to use guess, check, and
revise with mental math.
For example, try factoring the trinomial
x2 + 5x + 6.
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Factoring Trinomials of the Form x2 + bx + c
You need to find two numbers whose product is 6 and
whose sum is 5.
Using mental math, you can determine that the numbers
are 2 and 3.
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Example 1 – Finding the Greatest Common Factor
Factor the trinomial x2 + 5x – 6.
Solution:
You need to find two numbers whose product is –6 and
whose sum is 5.
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Example 2 – Finding the Greatest Common Factor
Factor the trinomial x2 – x – 6.
Solution:
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Factoring Trinomials of the Form x2 + bx + c
If you have trouble factoring a trinomial, it helps to make a
list of all the distinct pairs of factors and then check each
sum.
For instance, consider the trinomial
x2 – 5x – 24 = (x +
)(x –
).
Opposite signs
In this trinomial the constant term is negative, so you need
to find two numbers with opposite signs whose product is
–24 and whose sum is –5.
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Factoring Trinomials of the Form x2 + bx + c
Factors of –24
1, –24
–1, 24
2, –12
–2, 12
3, –8
–3, 8
4, –6
–4, 6
Sum
–23
23
–10
10
–5
5
–2
2
Correct choice
So, x2 – 5x – 24 = (x + 3)(x – 8).
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Factoring Trinomials of the Form x2 + bx + c
With experience, you will be able to narrow the list of
possible factors mentally to only two or three possibilities
whose sums can then be tested to determine the correct
factorization.
Here are some suggestions for narrowing the list.
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Factoring Trinomials in Two Variables
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Factoring Trinomials in Two Variables
The next example show how to factor trinomials of the form
x2 + bxy + cy2.
Note that this trinomial has two variables, x and y. However,
from the factorization
x2 + bxy + cy2 = x2 + (m + n)xy + mny2 = (x + my)(x + ny)
you can see that you still need to find two factors of c whose
sum is b.
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Example 5 – Factoring a Trinomial in Two Variables
Factor the trinomial x2 – xy – 12y2.
Solution:
You need to find two numbers whose product is –12 and
whose sum is –1.
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Example 6 – Factoring a Trinomial in Two Variables
Factor the trinomial x2 + 11xy + 10y2.
Solution:
You need to find two numbers whose product is 10and
whose sum is 11.
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Example 7 – Factoring a Trinomial in Two Variables
Factor the trinomial y2 – 6xy + 8x2.
Solution:
You need to find two numbers whose product is 8 and
whose sum is –6.
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Factoring Completely
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Factoring by Grouping
Some trinomials have a common monomial factor.
In such cases you should first factor out the common
monomial factor.
Then you can try to factor the resulting trinomial by the
methods of this section.
This “multiple-stage factoring process” is called factoring
completely. The trinomial below is completely factored.
2x2 – 4x – 6 = 2(x2 – 2x – 3)
= 2(x – 3)(x + 1)
Factor out common
monomial factor 2.
Factor trinomial.
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Example 8 – Factoring Completely
Factor the trinomial 2x2 – 12x + 10 completely.
Solution:
2x2 – 12x + 10 = 2(x2 – 6x + 5)
= 2(x – 5)(x – 1)
Factor out common
monomial factor 2.
Factor trinomial.
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Example 9 – Factoring Completely
Factor the trinomial 3x3 – 27x2 + 54x completely.
Solution:
3x3 – 27x2 + 54x = 3x(x2 – 9x + 18)
= 3x(x – 3)(x – 6)
Factor out common
monomial factor 3x.
Factor trinomial.
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Example 10 – Factoring Completely
Factor the trinomial 47y4 – 32y3 + 28y2 completely.
Solution:
4y4 – 32y3 + 28y2 = 4y2(y2 + 8y + 7)
= 4y2(y + 1)(y + 1)
Factor out common
monomial factor 4y2.
Factor trinomial.
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Example 11 – Geometry: Volume of an Open Box
An open box is to be made from a four-foot-by-six-foot sheet of metal
but cutting equal squares from the corners and turning up the sides.
The volume of the box can be modeled by V = 4x3 – 20x2 + 24x, 0 < x <
2.
a. Factor the trinomial that models the volume of the box. Use the
factored form to explain how the model was found.
b. Use a spreadsheet to approximate the size of the squares to be cut
from the corners so that the box has the maximum volume.
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Example 11 – Geometry: Volume of an Open Box
cont’d
Solution
a. 4x3 – 20x2 + 24x = 4x(x2 – 5x + 6)
= 4x(x – 3)(x – 2)
Factor out common
monomial factor 4x
Factored form
Because 4 = (–2)(–2), you can rewrite the factored form as
4x(x – 3)(x – 2) = x[(–2)(x – 3)][(–2)(x – 2)]
= x(6 – 2x)(4 – 2x)
= (6 – 2x)(4 – 2x)(x)
The model was found by multiplying the length, width, and
height of the box.
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Example 11 – Geometry: Volume of an Open Box
cont’d
b. From the spreadsheet below, you can see the maximum
volume of the box is about 8.45 cubic feet. This occurs
when the value of x is about 0.8 feet.
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Homework
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