6.2 Factor Trinomials
Topics covered in this section are:
- Factor trinomials of the form $x^{2}+bx+c$
- Factor trinomials of the form $x^{2}+bx+c$ using trial and error
- Factor trinomials of the form $ax^{2}+bx+c$ using the ‘ac’ method
- Factor using substitution
6.2.1 Factor Trinomials of the Form $x^{2}+bx+c$
You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication. To factor the trinomial means to start with the product, and end with the factors.
To figure out how we would factor a trinomial of the form $x^{2}+bx+c$, such as $x^{2}+5x+6$ and factor it to $(x+2)(x+3)$, let’s start with two general binomials of the form $(x+m)$ and $(x+n)$.
| $(x+m)(x+n)$ | |
| FOIL to find the product. | $x^{2}+mx+nx+n^{2}$ |
| Factor the GCF from the middle terms. | $x^{2}+(m+n)x+n^{2}$ |
| Our trinomial is of the form $x^{2}+bx+c$. |  |
This tell us that to factor a trinomial of the form $x^{2}+bx+c$, we need two factors $(x+m)$ and $(x+n)$ where the two numbers $m$ and $n$ multiply to $c$ and add to $b$.
Example 1
Factor: $x^{2}+11x+24$
Solution
| Step 1. Write the factor as two binomials with first terms $x$. | Write two sets of parentheses and put $x$ as the first term. | $x^{2}+11x+24$ $(x \ \ \ \ ) (x \ \ \ \ )$ |
| Step 2. Find two numbers $m$ and $n$ that multiply to $c$, $m \cdot n = c$ and add to $b$, $m+n=b$. | Find two numbers that multiply to $24$ and add to $11$ |  |
| Step 3. Use $m$ and $n$ as the last terms of the factors. | Use $3$ and $8$ as the last terms of the binomials. | $(x+3)(x+8)$ |
| Step 4. Check by multiplying the factors. | $(x+3)(x+8)$ $x^{2}+8x+3x+24$ $x^{2}+11x+24 \ \checkmark$ |
Let’s summarize the steps we used to find the factors.
HOW TO: Factor trinomials of the form $x^{2}+bx+c$.
- Write the factors as two binomials with first terms $x$.
$x^{2}+bx+c$
$(x \ \ \ \ )(x \ \ \ \ )$ - Find two numbers $m$ and $n$ that
- multiply to $c$, $m \cdot n=c$
- add to $b$, $m+n=b$
- Use $m$ and $n$ as the last terms of the factors.
$(x+m)(x+n)$ - Check by multiplying the factors.
In the first example, 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.
How do you get a positive product and a negative sum? We use two negative numbers.
Example 2
Factor: $y^{2}-11y+28$
Solution
Again, with the positive last term, $28$, and the negative middle term, $-11y$, we need two negative factors. Find two numbers that multiply to $28$ and add to $-11$.
| $y^{2}-11y+28$ | |
| Write the factors as two binomials with first terms $y$. | $(y \ \ \ \ )(y \ \ \ \ )$ |
| Find two numbers that: multiply to $28$ and add to $-11$. |
| Factors of $28$ | Sum of Factors |
|---|---|
| $-1, -28$ $-2, -14$ $-4, -7$ | $-1+(-28) = -29$ $-2+(-14)=-16$ $-4+(-7) = -11$ |
| Use $-4$, $-7$ as the last terms of the binomials. | $(y-4)(y-7)$ |
| Check: | $(y-4)(y-7)$ $y^{2}-7y-4y+28$ $y^{2}-11y+28 \ \checkmark$ |
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.
How do you get a negative product and a positive sum? We use one positive and one negative number.
When we factor trinomials, we must have the terms written in descending order—in order from highest degree to lowest degree.
Example 3
Factor: $2x+x^{2}-48$
Solution
| $2x+x^{2}-48$ | |
| First we put the terms in decreasing degree order. | $x^{2}+2x-48$ |
| Factors will be two binomials with first terms $x$. | $(x \ \ \ \ )(x \ \ \ \ )$ |
| Factors of $-48$ | Sum of Factors |
|---|---|
| $-1, 48$ $-2, 24$ $-3, 16$ $-4, 12$ $-6, 8$ | $-1+48 = 47$ $-2+24= 22$ $-3+16 = 13$ $-4+12=-8$ $-6+8=2$ |
| Use $-6$, $8$ as the last terms of the binomials. | $(x-6)(x+8)$ |
| Check: | $(x-6)(x+8)$ $x^{2}+8x-6x-48$ $x^{2}+2x-48 \ \checkmark$ |
Sometimes you’ll need to factor trinomials of the form $x^{2}+bxy+cy^{2}$ with two variables, such as $x^{2}+12xy+36y^{2}$. The first term, $x^{2}$, is the product of the first terms of the binomial factors, $x \cdot x$. The $y^{2}$ 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 How to Factor Trinomials.
Example 4
Factor: $r^{2}-8rs-9s^{2}$.
Solution
We need $r$ in the first term of each binomial and $s$ in the second terms. The last term of the trinomial is negative, so the factors must have opposite signs.
| $r^{2}-8rs-9s^{2}$ | |
| Note the the first terms are $r$, last terms contain $s$. | $(r \ \ \ \ s) (r \ \ \ \ s)$ |
| Find the numbers that multiply to $-9$ and add to $-8$. |
| Factors of $-9$ | Sum of Factors |
|---|---|
| $1, -9$ $-1, 9$ $3, -3$ | $1+(-9) = -8$ $-1+9= 8$ $3+(-3) = 0$ |
| Use $1$, $-9$ as coefficients of the last terms. | $(r+s)(r-9s)$ |
| Check: | $(r+s)(r-9s)$ $r^{2}-9rs+rs-9s^{2}$ $r^{2}-8rs-9s^{2} \ \checkmark$ |
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 5
Factor: $u^{2}-9uv-12v^{2}$.
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.
| $u^{2}-9uv-12v^{2}$ | |
| Note that the first terms are $u$, last terms contain $v$. | $(u \ \ \ \ v)(u \ \ \ \ v)$ |
| Find the numbers that multiply to $-12$ and add to $-9$. |
| Factors of $-12$ | Sum of Factors |
|---|---|
| $1, -12$ $-1, 12$ $2, -6$ $-2, 6$ $3, -4$ $-3, 4$ | $1+(-12) = -11$ $-1+12= 11$ $2+(-6) = -4$ $-2+6=4$ $3+(-4)=-1$ $-3+4=1$ |
Note there are no factor pairs that give us $-9$ as a sum. The trinomial is prime.
Let’s summarize the method we just developed to factor trinomials of the form $x^{2}+bx+c$.
STRATEGY FOR FACTORING TRINOMIALS OF THE FORM $x^{2}+bx+c$
When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.
$x^{2}+bx+c$
$(x+m)(x+n)$
When $c$ is positive, $m$ and $n$ have the same sign.
| $b$ positive | $b$ negative | |
| $m$, $n$ positive | $m$, $n$ negative | |
| $x^{2}+5x+6$ | $x^{2}-6x+8$ | |
| $(x+2)(x+3)$ | $(x-4)(x-2)$ | |
| same signs | same signs |
When $c$ is negative, $m$ and $n$ have opposite signs.
| $x^{2}+x-12$ | $x^{2}-2x-15$ | |
| $(x+4)(x-3)$ | $(x-5)(x+3)$ | |
| opposite signs | opposite signs |
Note that, in the case when $m$ and $n$ have opposite signs, the signs of the one with the larger absolute value matches the sign of $b$.
6.2.2 Factor Trinomials of the form $ax^{2}+bx+c$ using Trial and Error
Our next step is to factor trinomials whose leading coefficient is not $1$, trinomials of the form $ax^{2}+bx+c$.
Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes $1$ and you can factor it by the methods we’ve used so far. Let’s do an example to see how this works.
Example 6
Factor completely: $4x^{3}+16x^{2}-20x$.
Solution
| Is there a greatest common factor? | $4x^{3}+16x^{2}-20x$ |
| Yes, GCF $=4x$. Factor it. | $4x(x^{2}+4x-5)$ |
| Binomial, trinomial, or more than three terms? | |
| It is a trinomial. So “undo FOIL.” | $4x(x \ \ \ \ \ ) (x \ \ \ \ \ )$ |
| Use a table like the one show to find two numbers that multiply to $-5$ and add to $4$. |
| Factor of $-5$ | Sum of factors |
|---|---|
| $1, -5$ $-1, 5$ | $1+(-5)=-4$ $-1+5=4$ |
| Use $-1$, $5$. | $4x(x-1)(x+5)$ |
| Check: | $4x(x-1)(x+5)$ $4x(x^{2}+5x-x-5)$ $4x(x^{2}+4x-5)$ $4x^{3}+16x^{2}-20x \ \checkmark$ |
What happens when the leading coefficient is not $1$ and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.
Let’s factor the trinomial $3x^{2}+5x+2$.
From our earlier work, we expect this will factor into two binomials.
$3x^{2}+5x+2$
$( \ \ \ \ \ \ \ )( \ \ \ \ \ \ \ )$
We know the first terms of the binomial factors will multiply to give us $3x^{2}$. The only factors of $3x^{2}$ are $1x$, $3x$. We can place them in the binomials.
$(x \ \ \ \ \ )(3x \ \ \ \ \ )$
Check: Does $1x \cdot 3x=3x^{2}$? Yes.
We know the last terms of the binomials will multiply to $2$. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of $2$ are $1$, $2$. But we now have two cases to consider as it will make a difference if we write $1$, $2$ or $2$, $1$.
Which factors are correct? To decide that , we multiply the inner and outer terms.
Since the middle term of the trinomial is $5x$, the factors in the first case will work. Let’s use FOIL to check.
$(x+1)(3x+2)$
$3x^{2}+2x+3x+2$
$3x^{2}+5x+2 \ \checkmark$
Our result of the factoring is:
$3x^{2}+5x+2$
$(x+1)(3x+2)$
Example 7
Factor completely using trial and error: $3y^{2}+22y+7$.
Solution
| Step 1. Write the trinomial in descending order. | The trinomial is already in descending order. | $3y^{2}+22y+7$ |
| Step 2. Factor and GCF. | There is no GCF. | |
| Step 3. Find all the factor pairs of the first term. | The only factors of $3y^{2}$ are $1y$, $3y$. Since there is only one pair, we can put them in the parentheses. | $3y^{2}+22y+7$ $(y \ \ \ \ \ ) (3y \ \ \ \ \ \ )$ |
| Step 4. Find all the factor pairs of the third term. | The only factors of $y$ are $1$, $7$. | $(y+1)(3y+7)$ or $(y+7)(3y+1)$ |
| Step 5. Test all the possible combinations of the factors until the correct product is found. |  |  |
| Step 6. Check by multiplying. | $(y+7)(3y+1)$ $3y^{2}+y+21y+7$ $3y^{2}+22y+y \ \checkmark$ |
HOW TO: Factor trinomials of the form $ax^{2}+bx+c$ using trial and error.
- Write the trinomial in descending order of degrees as needed.
- Factor any GCF.
- Find all the factor pairs of the first term.
- Find all the factor pairs of the third term.
- Test all the possible combinations of the factors until the correct product is found.
- Check by multiplying.
Remember, when the middle term is negative and the last term is positive, the signs in the binomials must both be negative.
Example 8
Factor completely using trial and error: $6b^{2}-13b+5$.
Solution
| The trinomial is already in descending order. | $6b^{2}-13b+5$ |
| Find the factors of the first term. | $1b \cdot 6b$ $2b \cdot 3b$ |
| Find the factors of the last term. Consider the signs. Since the last term, $5$, is positive, its factors must both be positive or both be negative. The coefficient of the middle term is negative, so we use the negative factors. | $-1$, $-5$ |
Consider all the combinations of factors.
| Possible factors | Product |
|---|---|
| $(b-1)(6b-5)$ | $6b^{2}-11b+5$ |
| $(b-5)(6b-1)$ | $6b^{2}-31b+5$ |
| $(2b-1)(3b-5)$ | $6b^{2}-13b+5$ |
| $(2b-5)(3b-1)$ | $6b^{2}-17b+5$ |
| The correct factors are those whose product is the original trinomial. | $(2b-1)(3b-5)$ |
| Check by multiplying: | $(2b-1)(3b-5)$ $6b^{2}-10b-3b+5$ $6b^{2}-13b+5\ \checkmark$ |
When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.
Example 9
Factor completely using trial and error: $18x^{2}-37xy+15y^{2}$.
Solution
| The trinomial is already in descending order. | $18x^{2}-37xy+15y^{2}$ |
| Find the factors of the first term. | $1x \cdot 18x$ $2x \cdot 9x$ $3x \cdot 6x$ |
| Find the factors of the last term. Consider the signs. Since $15$ is positive and the coefficient of the middle term is negative, we use the negative factors. | $-1$, $-15$ $-3$, $-5$ |
Consider all the combinations of factors.
| The correct factors are those whose product is the original trinomial. | $(2x-3y)(9x-5y)$ |
| Check by multiplying: | $(2x-3y)(9x-5y)$ $18x^{2}-10xy-27xy+15y^{2}$ $18x^{2}-37xy+15y^{2} \ \checkmark$ |
Don’t forget to look for a GCF first and remember if the leading coefficient is negative, so is the GCF.
Example 10
Factor completely using trial and error: $-10y^{4}-55y^{3}-60y^{2}$.
Solution
| $-10y^{4}-55y^{3}-60y^{2}$ | |
| Notice the greatest common factor, so factor it first. | $-5y^{2}(2y^{2}+11y+12) |
| Write the factors of the first term. | $y \cdot 2y$ |
| Write the factors of the last term. | $1 \cdot 12$ $2 \cdot 6$ $3 \cdot 4$ |
Consider all the combinations.
| The correct factors are those whose product is the original trinomial. Remember to include the factor $-5y^{2}$. | $-5y^{2}(y+4)(2y+3)$ |
| Check by multiplying: | $-5y^{2}(y+4)(2y+3)$ $-5y^{2}(2y^{2}+3y+8y+12)$ $-10y^{4}-55y^{3}-60y^{2} \ \checkmark$ |
6.2.3 Factor Trinomials of the Form $ax^{2}+bx+c$ using the “ac” Method
Another way to factor trinomials of the form $ax^{2}+bx+c$ is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!
Example 11
Factor using the “ac” method: $6x^{2}+7x+2$.
Solution
| Step 1. Factor any GCF. | Is there a GCF? No! | $6x^{2}+7x+2$ |
| Step 2. Find the product of $ac$. | $a \cdot c$ $6 \cdot 2$ $12$ | $\textcolor{red}{ax^{2}+bx+x}$ $6x^{2}+7x+c$ |
| Step 3. Find two numbers $m$ and $n$ that: $\scriptsize{\text{Multiply to } ac, m \cdot n = a \cdot c}$ $\text{Add to } b, m+n=b$. | Find two number that’s multiply to $12$ and add to $7$. Both factors must be positive. $3 \cdot 4 = 12$ $3+4=7$ | |
| Step 4. Split the middle term using $m$ and $n$. $ax^{2}+bx+c$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bx \ \ \ \ \ $ $ax^{2} + \overbrace{mx+nx} +c$ | Rewrite $7x$ as $3x+4x$. It would also give us the same result if we used $4x+3x$. $6x^{2}+3x+4x+2$ is equal to $6x^{2}+7x+2$. We just split the middle term to get a more useful form. | $6x^{2}+7x+2$ $\underbrace{6x^{2}+3x} \underbrace{+4x+2}$ |
| Step 5. Factor by grouping. | $3x(2x+1)+2(2x+1)$ $(2x+1)(3x+2)$ | |
| Step 6. Check by multiplying the factors. | $(2x+1)(3x+2)$ $6x^{2}+4x+3x+2$ $6x^{2}+7x+2 \ \checkmark$ |
The “ac” method is summarized here.
HOW TO: Factor trinomials of the form $ax^{2}+bx+c$ using the “ac” method.
- Factor any GCF.
- Find the product $ac$.
- Find two numbers $m$ and $n$ that:
Multiply to $ac$ $ \ \ \ m \cdot n = ac$
Add to $b$ $ \ \ \ m+n=b$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ax^{2}+bx+c$ - Split the middle term using $m$ and $n$.
$ \ \ \ \ \ \ \ \ \ \ ax^{2}+mx+nx+c$ - Factor by grouping.
- Check by multiplying the factors.
Don’t forget to look for a common factor!
Example 12
Factor using the “ac” method: $10y^{2}-55y+70$
Solution
| Is there a GCF? | Yes. The GCF is $5$. | $10y^{2}-55y+70$ |
| Factor it. | $5(2y^{2}-11y+14)$ | |
| The trinomial inside the parentheses has a leading coefficient that is not $1$> | $\textcolor{red}{\ \ ax^{2} + bx \ + \ c}$ $5(2y^{2}-11y+14)$ | |
| Find the product of $ac$. | $ac=2 \cdot 14 = 28$ | |
| Find two number that multiply to $ac$ and add to $b$. | $(-4)(-7)=28$ $(-4)+(-7)=-11$ | |
| Split the middle term. | $5(\underbrace{2y^{2}-7y}\underbrace{-4y+14}$ | |
| Factor the trinomial by grouping. | $5(y(2y-7)-2(2y-7))$ $5(y-2)(2y-7)$ | |
| Check by multiplying all three factors. | $5(y-2)(2y-7)$ $5(2y^{2}-7y-4y+14)$ $5(2y^{2}-11y+14)$ $10y^{2}-55y+70 \ \checkmark$ |
6.2.4 Factor Using Substitution
Sometimes a trinomial does not appear to be in the $ax^{2}+bx+c$ form. However, we can often make a thoughtful substitution that will allow us to make it fit the $ax^{2}+bx+c$ form. This is called factoring by substitution. It is standard to use $u$ for the substitution.
In the $ax^{2}+bx+c$, the middle term has a variable, $x$, and its square, $x^{2}$, is the variable part of the first term. Look for this relationship as you try to find a substitution.
Example 13
Factor by substitution: $x^{4}-4x^{2}-5$.
Solution
The variable part of the middle term is $x^{2}$ and its square, $x^{4}$, is the variable part of the first term. (We know ($x^{2})^{2}=x^{4}$). If we let $u=x^{2}$, we can put our trinomial in the $ax^{2}+bx+c$ form we need to factor it.
| $x^{4}-4x^{2}-5$ | |
| Rewrite the trinomial to prepare for the substitution. | $(\textcolor{red}{x^{2}})^{2}-4(\textcolor{red}{x^{2}})-5$ |
| Let $u=x^{2}$ and substitute. | $\textcolor{red}{u}^{2}-4\textcolor{red}{u}-5$ |
| Factor the trinomial. | $(u+1)(u-5)$ |
| Replace $u$ with $x^{2}$. | $(\textcolor{red}{x^{2}}+1)(\textcolor{red}{x^{2}}-5)$ |
| Check: | $(x^{2}+1)(x^{2}-5)$ $x^{4}-5x^{2}+x^{2}-5$ $x^{4}-4x^{2}-5 \ \checkmark$ |
Sometimes the expression to be substituted is not a monomial.
Example 14
Factor by substitution: $(x-2)^{2}+7(x-2)+12$.
Solution
The binomial in the middle term, $(x-2)$ is squared in the first term. If we let $u=x-2$ and substitute, our trinomial will be in $ax^{2}+bx+c$ form.
| $(x-2)^{2}+7(x-2)+12$ | |
| Rewrite the trinomial to prepare for the substitution. | $\textcolor{red}{(x-2)}^{2}+7\textcolor{red}{(x-2)}+12$ |
| Let $u=x-2$ and substitute. | $\textcolor{red}{u}^{2}+7\textcolor{red}{u}+12$ |
| Factor the trinomial. | $(u+3)(u+4)$ |
| Replace $u$ with $x-2$. | $(\textcolor{red}{(x-2)}+3)(\textcolor{red}{(x-2)}+4)$ |
| Simplify inside the parentheses. | $(x+1)(x+2)$ |
This could also be factored by first multiplying out the $(x-2)^{2}$ and the $7(x-2)$ and then combining like terms and then factoring. Most students prefer the substitution method.
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- Revision and Adaption. Provided by: Minute Math. License: CC BY 4.0
CC Licensed Content, Shared Previously
- Marecek, L., & Mathis, A. H. (2020). Factor Trinomials 2e. OpenStax. https://openstax.org/books/intermediate-algebra-2e/pages/6-2-factor-trinomials. License: CC BY 4.0. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
