Fiveable
Fiveable

or

Log in

Find what you need to study


Light

Find what you need to study

Factoring Trinomials

4 min readdecember 10, 2021

J

Jaaziel Sandoval

J

Jaaziel Sandoval

Step by Step, The Easiest Way to Factor Trinomials

Introduction

When we’re factoring trinomials, where should we begin? Well, a trinomial is a polynomial that has three terms. It looks like this: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-OlNvdVqLMDmj.png?alt=media&token=3b143667-28c3-4593-8853-a05bd1ce0276

a’, ‘b’, and ‘c’ will be numbers and ‘x’ will be the variable, but it’s not always represented by an ‘x’.

  

Let’s practice with an example problem:

4x-8+7x^2+10=3x+6-2x^2

The first thing you will do to simplify this equation is move everything to one side. Do this by adding or subtracting the terms on both sides to have the equation equal zero.  

4x-8+7x^2+10-3x-6+2x^2=(3x-3x)(+6-6)(-2x^2+2x^2)

4x-8+7x^2+10-3x-6+2x^2=0

Now reorder the terms so that it mirrors the trinomial standard form.

7x2+2x2+4x-3x+10-6-8=0

Then, combine like terms.

9x^2+x-4=0

And just like that, the complicated-looking equation is now simple and easy!

Solutions

There are many ways to factor trinomials. Let’s look at a few solutions by factoring a trinomial three different ways! 

*Not all the ways we look at today will work for all trinomials you face, so try to learn different ways.*

Solution #1

The first way to factor trinomials is by using the quadratic formula. This article talks more about it, but the quadratic formula looks like this. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6AXiQ8CbhRk9.png?alt=media&token=196ef0c0-960f-461a-be9d-3b0f216be438

Basically, you’re going to plug in all the components from the earlier polynomial into the formula. Let’s practice! 

 2x^2+14x+24=0

First, let’s identify the components of the equation.

a=2

b=14

c=24

After identifying everything, let’s plug the numbers into the equation. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dVCShmTfpBCA.png?alt=media&token=de58559a-39ab-466b-b09a-3848d9aec9a3

Simplify. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-N7T27TnY4Opu.png?alt=media&token=014af3c6-275f-4725-bbbc-3fb489c5735c

Now what? What the heck does “”mean? This is the plus-minus symbol, and as the name suggests, you are going to add and subtract the equation. We can separate the two formulas to make things easier. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-EuLgBdlJ3m6K.png?alt=media&token=2966fbcd-205f-45f2-8bdc-78ab5e0409c9

Bam! You’ve got your answer. Notice how you got two answers? Don’t worry though, this is very common.

How can you check your work? Simply plug in either answer back into the original expression to check if the equation is true. Let’s see what that looks like. Let’s plug in -4 first. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-zpTd5ffjfStw.png?alt=media&token=6a5e6ff1-0253-4cbb-b4d6-eca174e331fe

The equation is true so we know one answer is correct. Let’s try -3 now.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-steVFLgjEceQ.png?alt=media&token=7236f484-0127-4f8a-8e37-7e9bb7e2a6e7

Since both equations are true, we know we got the right answer. 

Solution #2

Let’s look at another way to factor a trinomial.  

 2x^2+14x+24=0

First, identify your components. 

a=2

b=14

c=24

What you will do now is multiply ‘a’ times ‘c’, meaning 2 times 24. Then you will find a set of numbers that equal 24 and add up to “b”, in this case 14. 

To find the numbers that add up to 14 and multiply to 24, it may be beneficial to create a table like this one. Don’t forget negative numbers. 

Numbers that when combined add up to 24 

Sum

1

48

49

2

24

26

3

16

19

4

12

16

6

8

14

-1

-48

-49

-2

-24

-26

-3

-16

-19

-4

-12

-16

-6

-8

-14

 The sets of numbers that add up to 14 and multiply to 24 are 6 and 8. Now you will separate ‘b’ a.k.a. 14 into two parts, the two parts are 6 and 8. The new expression will look like this. 

 2x^2+6x+8x+24=0

Now, you will separate the equation into two parts. 

2x^2+6x=0     8x+24=0

Then, you will factor out the two equations. Let’s do this one first. 

2x^2+6x=0

You will take out what both parts have in common. In this case, both parts contain a “2x”. Let’s take it out. 

2x (x+3)=0

Let’s do the second one. 

8x+24=0

Both of the parts have an 8 in common, so let’s take it out. 

8(x+3)=0

Let’s take a look at both factored equations now. 

2x (x+3)=0   8(x+3)=0

Notice that both equations have (x+3) in common. That will be one of the answers. The other answer will be the combination of the parts taken out from the two equations. Let’s see what that looks like.

(x+3)=0      (2x+8)=0

Now let’s solve for x, which will give us our two answers. 

(x+3)=0      (2x+8)=0

x+3-3=0-3      2x+8-8=0-8

x=-3             2x/2=-8/2

x=-3     x=-4

Piece of cake!

Solution #3

The third way to factor out trinomials is simplifying the equation by taking out the greatest common factor. 

 2x^2+14x+24=0

You can take out a 2. 

2(x^2+7x+12)=0

By dividing both sides by 2, we can dump it.

(x^2+7x+12)=0

Now what we will do is similar to the 2 second solution. You must find a set of numbers that add up to ‘b’ and multiply to ‘c’. In this case, ‘b’ is 7 and ‘c’ is 12, so you must find a set of numbers that add up to 7 and multiply to 12. Let’s make a table. 

Numbers that when combined add up to 12 

Sum

1

12

13

2

6

8

3

4

7

-1

-12

-13

-2

-6

-8

-3

-4

-7

The sets of numbers that add up to 12 and multiply to 7 are 3 and 4.

Now you’ll just add these numbers into this expression. 

(x+_)=0

Let’s see what that looks like. 

(x+3)=0     (x+4)=0

Now just solve for x. 

(x+3)=0      (x+4)=0

x+3-3=0-3      x+4-4=0-4

x=-3     x=-4

Easy peasey, lemon squeezy. Notice that this way is the easiest, but won’t work with every trinomial. 

Closing 

Factoring trinomials is all about following steps, once you’ve mastered these steps no trinomial will get past you!

Factoring Trinomials

4 min readdecember 10, 2021

J

Jaaziel Sandoval

J

Jaaziel Sandoval

Step by Step, The Easiest Way to Factor Trinomials

Introduction

When we’re factoring trinomials, where should we begin? Well, a trinomial is a polynomial that has three terms. It looks like this: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-OlNvdVqLMDmj.png?alt=media&token=3b143667-28c3-4593-8853-a05bd1ce0276

a’, ‘b’, and ‘c’ will be numbers and ‘x’ will be the variable, but it’s not always represented by an ‘x’.

  

Let’s practice with an example problem:

4x-8+7x^2+10=3x+6-2x^2

The first thing you will do to simplify this equation is move everything to one side. Do this by adding or subtracting the terms on both sides to have the equation equal zero.  

4x-8+7x^2+10-3x-6+2x^2=(3x-3x)(+6-6)(-2x^2+2x^2)

4x-8+7x^2+10-3x-6+2x^2=0

Now reorder the terms so that it mirrors the trinomial standard form.

7x2+2x2+4x-3x+10-6-8=0

Then, combine like terms.

9x^2+x-4=0

And just like that, the complicated-looking equation is now simple and easy!

Solutions

There are many ways to factor trinomials. Let’s look at a few solutions by factoring a trinomial three different ways! 

*Not all the ways we look at today will work for all trinomials you face, so try to learn different ways.*

Solution #1

The first way to factor trinomials is by using the quadratic formula. This article talks more about it, but the quadratic formula looks like this. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6AXiQ8CbhRk9.png?alt=media&token=196ef0c0-960f-461a-be9d-3b0f216be438

Basically, you’re going to plug in all the components from the earlier polynomial into the formula. Let’s practice! 

 2x^2+14x+24=0

First, let’s identify the components of the equation.

a=2

b=14

c=24

After identifying everything, let’s plug the numbers into the equation. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dVCShmTfpBCA.png?alt=media&token=de58559a-39ab-466b-b09a-3848d9aec9a3

Simplify. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-N7T27TnY4Opu.png?alt=media&token=014af3c6-275f-4725-bbbc-3fb489c5735c

Now what? What the heck does “”mean? This is the plus-minus symbol, and as the name suggests, you are going to add and subtract the equation. We can separate the two formulas to make things easier. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-EuLgBdlJ3m6K.png?alt=media&token=2966fbcd-205f-45f2-8bdc-78ab5e0409c9

Bam! You’ve got your answer. Notice how you got two answers? Don’t worry though, this is very common.

How can you check your work? Simply plug in either answer back into the original expression to check if the equation is true. Let’s see what that looks like. Let’s plug in -4 first. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-zpTd5ffjfStw.png?alt=media&token=6a5e6ff1-0253-4cbb-b4d6-eca174e331fe

The equation is true so we know one answer is correct. Let’s try -3 now.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-steVFLgjEceQ.png?alt=media&token=7236f484-0127-4f8a-8e37-7e9bb7e2a6e7

Since both equations are true, we know we got the right answer. 

Solution #2

Let’s look at another way to factor a trinomial.  

 2x^2+14x+24=0

First, identify your components. 

a=2

b=14

c=24

What you will do now is multiply ‘a’ times ‘c’, meaning 2 times 24. Then you will find a set of numbers that equal 24 and add up to “b”, in this case 14. 

To find the numbers that add up to 14 and multiply to 24, it may be beneficial to create a table like this one. Don’t forget negative numbers. 

Numbers that when combined add up to 24 

Sum

1

48

49

2

24

26

3

16

19

4

12

16

6

8

14

-1

-48

-49

-2

-24

-26

-3

-16

-19

-4

-12

-16

-6

-8

-14

 The sets of numbers that add up to 14 and multiply to 24 are 6 and 8. Now you will separate ‘b’ a.k.a. 14 into two parts, the two parts are 6 and 8. The new expression will look like this. 

 2x^2+6x+8x+24=0

Now, you will separate the equation into two parts. 

2x^2+6x=0     8x+24=0

Then, you will factor out the two equations. Let’s do this one first. 

2x^2+6x=0

You will take out what both parts have in common. In this case, both parts contain a “2x”. Let’s take it out. 

2x (x+3)=0

Let’s do the second one. 

8x+24=0

Both of the parts have an 8 in common, so let’s take it out. 

8(x+3)=0

Let’s take a look at both factored equations now. 

2x (x+3)=0   8(x+3)=0

Notice that both equations have (x+3) in common. That will be one of the answers. The other answer will be the combination of the parts taken out from the two equations. Let’s see what that looks like.

(x+3)=0      (2x+8)=0

Now let’s solve for x, which will give us our two answers. 

(x+3)=0      (2x+8)=0

x+3-3=0-3      2x+8-8=0-8

x=-3             2x/2=-8/2

x=-3     x=-4

Piece of cake!

Solution #3

The third way to factor out trinomials is simplifying the equation by taking out the greatest common factor. 

 2x^2+14x+24=0

You can take out a 2. 

2(x^2+7x+12)=0

By dividing both sides by 2, we can dump it.

(x^2+7x+12)=0

Now what we will do is similar to the 2 second solution. You must find a set of numbers that add up to ‘b’ and multiply to ‘c’. In this case, ‘b’ is 7 and ‘c’ is 12, so you must find a set of numbers that add up to 7 and multiply to 12. Let’s make a table. 

Numbers that when combined add up to 12 

Sum

1

12

13

2

6

8

3

4

7

-1

-12

-13

-2

-6

-8

-3

-4

-7

The sets of numbers that add up to 12 and multiply to 7 are 3 and 4.

Now you’ll just add these numbers into this expression. 

(x+_)=0

Let’s see what that looks like. 

(x+3)=0     (x+4)=0

Now just solve for x. 

(x+3)=0      (x+4)=0

x+3-3=0-3      x+4-4=0-4

x=-3     x=-4

Easy peasey, lemon squeezy. Notice that this way is the easiest, but won’t work with every trinomial. 

Closing 

Factoring trinomials is all about following steps, once you’ve mastered these steps no trinomial will get past you!



© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.