Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

7.2: Factor Quadratic Trinomials with Leading Coefficient 1

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives

By the end of this section, you will be able to:

  • Factor trinomials of the form x2+bx+c
  • Factor trinomials of the form x2+bxy+cy2
BE PREPARED

Before you get started, take this readiness quiz.

  1. Multiply: (x+4)(x+5).
    If you missed this problem, review Example 6.3.31.
  2. Simplify: (a) 9+(6) (b) 9+6.
    If you missed this problem, review Example 1.4.18.
  3. Simplify: (a) 9(6) (b) 9(6).
    If you missed this problem, review Example 1.5.1.
  4. Simplify: (a) |5| (b) |3|.
    If you missed this problem, review Example 1.4.2.

Factor Trinomials of the Form x2+bx+c

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.

This figure shows the steps of multiplying the factors (x + 2) times (x + 3). The multiplying is completed using FOIL to demonstrate. The first term is x squared and is below F. The second term is 3 x below “O”. The third term is 2 x below “I”. The fourth term is 6 below L. The simplified product is then given as x 2 plus 5 x + 6.

To factor the trinomial means to start with the product, x2+5x+6, and end with the factors, (x+2)(x+3). You need to think about where each of the terms in the trinomial came from.

The first term came from multiplying the first term in each binomial. So to get x2 in the product, each binomial must start with an x.

x2+5x+6(x)(x)

The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6.

What two numbers multiply to 6?

The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?

Consider the middle term. It came from adding the outer and inner terms.

So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in Table 7.2.1—the table will be very helpful when you work with numbers that can be factored in many different ways.

Factors of 6 Sum of factors
1,6 1+6=7
2,3 2+3=5

Table 7.2.1

We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of x2+5x+6. They are (x+2)(x+3).

x2+5x+6 product (x+2)(x+3) factors 

You should check this by multiplying.

Looking back, we started with x2+5x+6, which is of the form x2+bx+c, where b=5 and c=6. We factored it into two binomials of the form (x+m) and (x+n).

x2+5x+6x2+bx+c(x+2)(x+3)(x+m)(x+n)

To get the correct factors, we found two numbers m and n whose product is c and sum is b.

Example 7.2.1: HOW TO FACTOR TRINOMIALS OF THE FORM x2+bx+c

Factor: x2+7x+12

Solution

This table gives the steps for factoring x squared + 7 x + 12. The first row states the first step “write the factors as two binomials with first terms x”. In the second column of the first row it states, “write two sets of parentheses and put x as the first term”. In the third column, it has the expression x squared + 7 x +12. Below the expression are two sets of parentheses with x as the first term.The second row states the second step “find two numbers m and n that multiply to c, m times n = c and add to b, m + n = b”. In the second column of the second row are the factors of 12 and their sums. 1,12 with sum 1 + 12 = 13. 2, 6 with sum 2 + 6 =8. 3, 4 with sum 3 + 4 = 7.The third row states “use m and n as the last terms of the factors”. The second column states “use 3 and 4 as the last terms of the binomials”. The third column in this row has the product (x + 3)(x + 4).In the fourth row the statement is “check by multiplying the factors”. The product of (x + 3)(x +4) is shown to be x 2 + 7 x + 12.

Try It 7.2.2

Factor: x2+6x+8

Try It 7.2.3

Factor: y2+8y+15

Let’s summarize the steps we used to find the factors.

HOW TO

Factor trinomials of the form x2+bx+c.

Step 1. Write the factors as two binomials with first terms x: (x)(x)

Step 2. Find two numbers m and n that
 Multiply to c, mn=c
 Add to b, m+n=b

Step 3. Use m and n as the last terms of the factors:(x+m)(x+n)

Step 4. Check by multiplying the factors.

Example 7.2.4

Factor: u2+11u+24

Solution

Notice that the variable is u, so the factors will have first terms u.

u2+11u+24 Write the factors as two binomials with first terms u . (u)(u) Find two numbers that: multiply to 24 and add to 11.

Factors of 24 Sum of factors
1,24 1+24=25
2,12 2+12=14
3,8 3+8=11*
4,6 4+6=10

 Use 3 and 8 as the last terms of the binomials. (u+3)(u+8) Check. (u+3)(u+8)u2+3u+8u+24u2+11u+24v

Try It 7.2.5

Factor: q2+10q+24

Try It 7.2.6

Factor: t2+14t+24

Example 7.2.7

Factor: y2+17y+60

Solution

y2+17y+60 Write the factors as two binomials with first terms y. (y)(y)

Find two numbers that multiply to 60 and add to 17.

Factors of 60 Sum of factors
1,60 1+60=61
2,30 2+30=32
3,20 3+20=23
4,15 4+15=19
5,12 5+12=17*
6,10  
 Use 5 and 12 as the last terms. (y+5)(y+12) Check.(y+5)(y+12)(y2+12y+5y+60)(y2+17y+60)
Try It 7.2.8

Factor: x2+19x+60

Try It 7.2.9

Factor: v2+23v+60

Factor Trinomials of the Form x2 + bx + c with b Negative, c Positive

In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.

Remember: To get a negative sum and a positive product, the numbers must both be negative.

Again, think about FOIL and where each term in the trinomial came from. Just as before,

  • the first term, x2, comes from the product of the two first terms in each binomial factor, x and y;
  • the positive last term is the product of the two last terms
  • the negative middle term is the sum of the outer and inner terms.

How do you get a positive product and a negative sum? With two negative numbers.

Example 7.2.10

Factor: t211t+28

Solution

Again, with the positive last term, 28, and the negative middle term, −11t, we need two negative factors. Find two numbers that multiply 28 and add to −11.

t211t+28Write the factors as two binomials with first terms t(t)(t)

Find two numbers that: multiply to 28 and add to −11.

Factors of 28 Sum of factors
−1,−28 −1+(−28)=−29
−2,−14 −2+(−14)=−16
−4,−7 4+(7)=11
 Use 4,7 as the last terms of the binomials. (t4)(t7) Check. (t4)(t7)t27t4t+28t211t+28
Try It 7.2.11

Factor: u29u+18

Try It 7.2.12

Factor: y216y+63

Factor Trinomials of the Form x2+bx+c with c Negative

Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.

Remember: To get a negative product, the numbers must have different signs.

Example 7.2.13

Factor: z2+4z5

Solution

To get a negative last term, multiply one positive and one negative. We need factors of −5 that add to positive 4.

Factors of −5 Sum of factors
1,−5 1+(−5)=−4
−1,5 −1+5=4*

Notice: We listed both 1,−5 and −1,5 to make sure we got the sign of the middle term correct.

z2+4z5 Factors will be two binomials with first terms z. (z)(z) Use 1,5 as the last terms of the binomials. (z1)(z+5) Check. (z1)(z+5)z2+5z1z5z2+4z5

Try It 7.2.14

Factor: h2+4h12

Answer

(h2)(h+6)

Try It 7.2.15

Factor: :22+k20

Answer

(k4)(k+5)

Let’s make a minor change to the last trinomial and see what effect it has on the factors.

Example 7.2.16

Factor: z24z5

Solution

Solution

This time, we need factors of −5 that add to −4.

Factors of −5 Sum of factors
1,−5 1+(−5)=−4*
−1,5 −1+5=4

z24z5 Factors will be two binomials with first terms z. (z)(z) Use 1,5 as the last terms of the binomials. (z+1)(z5) Check. (z+1)(z5)z25z+1z5z24z5

Try It 7.2.17

Factor: x24x12

Answer

(x+2)(x6)

Try It 7.2.18

Factor: y2y20

Answer

(y+4)(y5)

Example 7.2.19

Factor: q22q15

Solution

q22q15 Factors will be two binomials with first terms q. (q)(q) You can use 3,5 as the last terms of the binomials. (q+3)(q5)

Factors of −15 Sum of factors
1,−15 1+(−15)=−14
−1,15 −1+15=14
3,−5 3+(−5)=−2*
−3,5

 Check. (q+3)(q5)q25q+3z15q22q15

Try It 7.2.20

Factor: r23r40

Answer

(r+5)(r8)

Try It 7.2.21

Factor: s23s10

Answer

(s+2)(s5)

Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.

Example 7.2.22

Factor: y26y+15

Solution

y26y+15 Factors will be two binomials with first (y)(y) terms y. 

Factors of 15 Sum of factors
−1,−15 −1+(−15)=−16
−3,−5 −3+(−5)=−8

As shown in the table, none of the factors add to −6; therefore, the expression is prime.

Try It 7.2.23

Factor: m2+4m+18

Answer

prime

Try It 7.2.24

Factor: n210n+12

Answer

prime

Example 7.2.25

Factor: 2x+x248

Solution

2x+x248 First we put the terms in decreasing degree order. x2+2x48 Factors will be two binomials with first terms x . (x)(x)

As shown in the table, you can use −6,8 as the last terms of the binomials.

(x6)(x+8)

Factors of −48 Sum of factors
−1,48 −1+48=47
−2,24
−3,16
−4,12
−6,8
−2+24=22
−3+16=13
−4+12=8
−6+8=2

 Check. (x6)(x+8)x26q+8q48x2+2x48

Try It 7.2.26

Factor: 9m+m2+18

Answer

(m+3)(m+6)

Try It 7.2.27

Factor: 7n+12+n2

Answer

(n3)(n4)

Let’s summarize the method we just developed to factor trinomials of the form x2+bx+c

Note

When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

x2+bx+c(x+m)(x+n)

When c is positive, m and n have the same sign.

 b positive  b negative m,n positive m,n negative x2+5x+6x26x+8(x+2)(x+3)(x4)(x2) same signs  same signs 

When c is negative, m and n have opposite signs.

x2+x12x22x15(x+4)(x3)(x5)(x+3) opposite signs  opposite signs 

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

Factor Trinomials of the Form x2 + bxy + cy2

Sometimes you’ll need to factor trinomials of the form x2+bxy+cy2 with two variables, such as x2+12xy+36y2. The first term, x2, is the product of the first terms of the binomial factors, xx. The y2 in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.

Example 7.2.28

Factor: x2+12xy+36y2

Solution

x2+12xy+36y2 Note that the first terms are x, last terms (xy)(xy) contain y

Find the numbers that multiply to 36 and add to 12.

Factors of 36 Sum of factors
1, 36 1+36=37
2, 18 2+18=20
3, 12 3+12=15
4, 9 4+9=13
6, 6 6+6=12*

 Use 6 and 6 as the coefficients of the last terms. (x+6y)(x+6y) Check your answer. 

(x+6y)(x+6y)x2+6xy+6xy+36y2x2+12xy+36y2

Try It 7.2.29

Factor: u2+11uv+28v2

Answer

(u+4v)(u+7v)

Try It 7.2.30

Factor: x2+13xy+42y2

Answer

(x+6y)(x+7y)

Example 7.2.31

Factor: r28rs9s2

Solution

We need r in the first term of each binomial and s in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

r28rs9s2 Note that the first terms are r, last terms contain s(rs)(rs)

Factors of −9 Sum of factors
1,−9 1+(−9)=−8*
−1,9 −1+9=8
3,−3 3+(−3)=0

 Use 1,9 as coefficients of the last terms. (r+s)(r9s) Check your answer. 

(r9s)(r+s)r2+rs9rs9s2r28rs9s2

Try It 7.2.32

Factor: a211ab+10b2

Answer

(ab)(a10b)

Try It 7.2.33

Factor: m213mn+12n2

Answer

(mn)(m12n)

Example 7.2.34

Factor: u29uv12v2

Solution

We need u in the first term of each binomial and v in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

u29uv12v2 Note that the first terms are u, last terms contain v(uv)(uv)

Find the numbers that multiply to −12 and add to −9.

Factors of −12 Sum of factors
1,−12 1+(−12)=−11
−1,12 −1+12=11
2,−6 2+(−6)=−4
−2,6 −2+6=4
3,−4 3+(−4)=−1
−3,4 −3+4=1

Note there are no factor pairs that give us −9 as a sum. The trinomial is prime.

Try It 7.2.35

Factor: x27xy10y2

Answer

prime

Try It 7.2.36

Factor: p2+15pq+20q2

Answer

prime

Key Concepts

  • Factor trinomials of the form x2+bx+c
    1. Write the factors as two binomials with first terms x: (x)(x)
    2. Find two numbers m and n that
      Multiply to c, mn=c
      Add to b, m+n=b
    3. Use m and n as the last terms of the factors: (x+m)(x+n).
    4. Check by multiplying the factors.

This page titled 7.2: Factor Quadratic Trinomials with Leading Coefficient 1 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?