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7.2: Factoring trinomials of the form x² + bx + c

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Factoring with three terms, or trinomials, is the most important technique, especially in further algebra. Since factoring is a product of factors, we first look at multiplying to develop the process of factoring trinomials.

Factoring Trinomials of the Form x2+bx+c

If we multiply (x+p)(x+q), we would get

x2+px+qx+pqx2+(p+q)x+pq

Notice the two factors of the last coefficient must add up to be the middle coefficient, i.e.,

pq=c and p+q=b

Hence, if we can find two numbers whose sum is b and that multiply to c, then we can split the middle term and factor by grouping.

Steps for factoring trinomials of the form x2+bx+c

Step 1. Find two numbers, p and q, whose sum is b and product is c.

Step 2. Rewrite the expression so that the middle term is split into two terms, p and q.

Step 3. Factor by grouping.

Step 4. Verify the factored form by finding the product.

Example 7.2.1

Factor: x2+9x+18

Solution

First we identify b=9 and c=18. We ask ourselves,“ What two numbers multiply to 18 that add up to 9?”

Step 1. Find two numbers whose sum is 9 and product is 18:

Table 7.2.1
p and q Product Sum
2,9 18 11
3,6 18 9
1,18 18 19

We can see from the table that 3 and 6 are the two numbers whose product is 18 and sum is 9. We use these two numbers in Step 2.

Step 2. Rewrite the expression so that the middle term is split into two terms, 3x and 6x: x2+9x+18x2+3x+6xsum is 9x+18

Step 3. Factor by grouping. x2+6x+3x+18Group the first two terms and the last two terms(x2+3x)+(6x+18)Factor x from the first group and 6from the second groupx(x+3)+6(x+3)Factor the GCF (x+3)(x+3)(x+6)Factored form

Step 4. Verify the factored form by finding the product: (x+3)(x+6)FOILx2+6x+3x+18Combine like termsx2+9x+18 Original expression

Thus, the factored form is (x+3)(x+6).

Example 7.2.2

Factor: x24x+3

Solution

First we identify b=4 and c=3. We ask ourselves,“ What two numbers multiply to 3 that add up to 4?”

Step 1. Find two numbers whose sum is 4 and product is 3:

Table 7.2.2
p and q Product Sum
1,3 3 4
1,3 3 4

We can see from the table that 1 and 3 are the two numbers whose product is 3 and sum is 4. We use these two numbers in Step 2.

Step 2. Rewrite the expression so that the middle term is split into two terms, 1x and 3x: x24x+3x21x3xsum is 4x+3

Step 3. Factor by grouping. x21x3x+3Group the first two terms and the last two terms(x21x)+(3x+3)Factor x from the first group and 3from the second groupx(x1)3(x1)Factor the GCF (x1)(x1)(x3)Factored form

Step 4. Verify the factored form by finding the product: (x1)(x3)FOILx21x3x+3Combine like termsx24x+3 Original expression

Thus, the factored form is (x1)(x+3).

Example 7.2.3

Factor: x28x20

Solution

First we identify b=8 and c=20. We ask ourselves,“ What two numbers multiply to 20 that add up to 8?”

Step 1. Find two numbers whose sum is 8 and product is 20:

Table 7.2.3
p and q Product Sum
4,5 20 1
4,5 10 1
2,10 20 8
2,10 20 8
1,20 20 19
1,20 20 19

We can see from the table that 2 and 10 are the two numbers whose product is 20 and sum is 8. We use these two numbers in Step 2.

Step 2. Rewrite the expression so that the middle term is split into two terms, 2x and 10x: x28x20x2+2x10xsum is 8x20

Step 3. Factor by grouping. x2+2x10x20Group the first two terms and the last two terms(x2+2x)+(10x20)Factor x from the first group and 10from the second groupx(x+2)10(x+2)Factor the GCF (x+2)(x+2)(x10)Factored form

Step 4. Verify the factored form by finding the product: (x+2)(x10)FOILx2+2x10x20Combine like termsx28x20 Original expression

Thus, the factored form is (x+2)(x+10).

Example 7.2.4

Factor: a29ab+14b2

Solution

First we identify b=9 and c=14. We ask ourselves,“ What two numbers multiply to 14 that add up to 9?”

Step 1. Find two numbers whose sum is 9 and product is 14:

Table 7.2.4
p and q Product Sum
2,7 14 9
2,7 14 9
1,14 14 15
1,14 14 15

We can see from the table that 2 and 7 are the two numbers whose product is 14 and sum is 9. We use these two numbers in Step 2.

Step 2. Rewrite the expression so that the middle term is split into two terms, 2ab and 7ab: a29ab+14b2a22ab7absum is 9ab+14b2

Step 3. Factor by grouping. a22ab7ab+14b2Group the first two terms and the last two terms(a22ab)+(7ab+14b2)Factor a from the first group and 7bfrom the second groupa(a2b)7b(a2b)Factor the GCF (a2b)(a2b)(a7b)Factored form

Step 4. Verify the factored form by finding the product: (a2b)(a7b)FOILa22ab7ab+14b2Combine like termsa29ab+14b2 Original expression

Thus, the factored form is (a2b)(a7b).

Note

There is a shortcut for factoring expressions of the type x2+bx+c. Once we identify the two numbers, p and q, whose product is c and sum is b, we can see these two numbers are the numbers in the factored form, i.e., (x+p)(x+q). We can use this shortcut only when the coefficient of x2 is 1. (We discuss when the coefficient is a number other than 1 in the next section.)

Example 7.2.5

Factor: x27x18

Solution

First we identify b=7 and c=18. We ask ourselves,“ What two numbers multiply to 18 that add up to 7?”

Table 7.2.5
p and q Product Sum
2,9 18 7
2,9 18 7
1,18 18 17
1,18 18 17

We can see from the table that 2 and 9 are the two numbers whose product is 18 and sum is 7. We use these two numbers to rewrite the expression in factored form:

x27x18(x+2)(x9)

We can always verify the factored form by multiplying and obtaining the original expression.

Example 7.2.6

Factor: m2mn30n2

Solution

First we identify b=1 and c=30. We ask ourselves,“ What two numbers multiply to 30 that add up to 1?”

Table 7.2.6
p and q Product Sum
2,15 30 13
2,15 30 13
5,6 30 1
5,6 30 1
1,30 30 29
1,30 30 29

We can see from the table that 5 and 6 are the two numbers whose product is 30 and sum is 1. We use these two numbers to rewrite the expression in factored form:

m2mn30n2(m+5n)(m6n)

We can always verify the factored form by multiplying and obtaining the original expression.

Example 7.2.7

Factor: x2+2x+6

Solution

First we identify b=2 and c=6. We ask ourselves,“ What two numbers multiply to 6 that add up to 2?”

Table 7.2.7
p and q Product Sum
2,3 6 5
2,3 6 5
1,6 6 7
1,6 6 7

We can see from the table that there aren’t any factors of 6 whose sum is 2. We only obtain sums with 5 and 7’s. In this case, we call this trinomial not factorable, or better yet, the trinomial is prime.

Note

If a trinomial (or polynomial) is not factorable, then we say we the trinomial is prime.

Factoring Trinomials of the Form x2+bx+c with a Greatest Common Factor

Factoring the GCF is always the first step in factoring expressions. If all terms have a common factor, we, first, factor the GCF and then factor as usual.

Example 7.2.8

Factor: 3x224x+45

Solution

Notice all three terms have a common factor of 3. We factor a 3 first, then factor as usual.

3x224x+45Factor the GCF3(x28x+15)

Next, we only concentrate on the expression in the parenthesis. What two numbers multiply to 15 that add up to 8?

Table 7.2.8
p and q Product Sum
3,5 15 8
3,5 15 8

We can see from the table that 3 and 5 are the two numbers whose product is 15 and sum is 8. We use these two numbers to rewrite the expression in factored form:

3x224x+43(x3)(x5)

We can always verify the factored form by multiplying and obtaining the original expression.

Note

Students tend to forget to write the GCF in the final answer. Be sure to always include the GCF in the final factored form.

Also, to factor completely, it is required the GCF is factored out of the expression. If not, then the expression is not factored completely.

Note

The first person to use letters for unknown values was Francois Vieta in 1591 in France. He used vowels to represent variables for solving, just as codes used letters to represent an unknown message.

Factoring Trinomials of the Form x2+bx+c Homework

Factor completely.

Exercise 7.2.1

p2+17p+72

Exercise 7.2.2

n29n+8

Exercise 7.2.3

x29x10

Exercise 7.2.4

b2+12b+32

Exercise 7.2.5

x2+3x70

Exercise 7.2.6

n28n+15

Exercise 7.2.7

p2+15p+54

Exercise 7.2.8

n215n+56

Exercise 7.2.9

u28uv+15v2

Exercise 7.2.10

m2+2mn8n2

Exercise 7.2.11

x211xy+18y2

Exercise 7.2.12

x2+xy12y2

Exercise 7.2.13

x2+4xy12y2

Exercise 7.2.14

5a2+60a+100

Exercise 7.2.15

6a2+24a192

Exercise 7.2.16

6x2+18xy+12y2

Exercise 7.2.17

6x2+96xy+378y2

Exercise 7.2.18

x2+x72

Exercise 7.2.19

x2+x30

Exercise 7.2.20

x2+13x+40

Exercise 7.2.21

b217b+70

Exercise 7.2.22

x2+3x18

Exercise 7.2.23

a26a27

Exercise 7.2.24

p2+7p30

Exercise 7.2.25

m215mn+50n2

Exercise 7.2.26

m23mn40n2

Exercise 7.2.27

x2+10xy+16y2

Exercise 7.2.28

u29uv+14v2

Exercise 7.2.29

x2+14xy+45y2

Exercise 7.2.30

4x2+52x+168

Exercise 7.2.31

5n245n+40

Exercise 7.2.32

5v2+20v25

Exercise 7.2.33

5m2+30mn90n2

Exercise 7.2.34

6m236mn162n2


This page titled 7.2: Factoring trinomials of the form x² + bx + c is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

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