Factoring is an important skill going into calculus. You will be doing a LOT of factoring in calculus. Master these techniques now and you will do much better in calculus. Many students struggle with calculus because their algebra skills are lacking, including factoring.
Recommended Books on Amazon (affiliate links)  

Join Amazon Student  FREE TwoDay Shipping for College Students 
There are lots of factoring techniques and, unfortunately, there is no one technique that works in all cases. So you need to learn them well. You will find several that you will use a lot. Learn those first. Then keep a bookmark for this page and come back here when you need to remind yourself of the other techniques.
The major techniques are labeled as primary. Learn these well. The secondary techniques are usually special cases and you will use them but not as often as the primary techniques.
However, there are a few steps that will help you get started with all equations.
Steps
1. Rewrite the polynomial with terms in order, highest term on the left. Of course, it would still work if you wrote them in reverse order but most of the time you will see your instructor and textbook written left to right, highest power on the left. So we suggest that you do the same.
2. Look for obvious common factors in all terms, like \(x\) or constants. You do not need to see all of them at the same time. If you see one term, factor it out and then look for more until you think they are all factored out. Here is an example.
Completely factor this polynomial.
\( 3x^2 + 6x \)
Problem Statement 

Factor \( 3x^2 + 6x \)
Final Answer 

\( 3x^2 + 6x = 3x(x + 2) \)
Problem Statement 

Factor \( 3x^2 + 6x \)
Solution 

First, we will break each factor down into individual terms. 
\( 3x^2 = 3 \cdot x \cdot x\) 
\( 6x = 2 \cdot 3 \cdot x \) 
Now we can see the common factors in each term. 
In both terms we have a 3, so let's factor out a 3 first. 
\( 3 \cdot x \cdot x + 2 \cdot 3 \cdot x = 3( x \cdot x + 2 \cdot x) \) 
Okay, so now let's look at what is left inside the parentheses. Notice that in each of those terms, we have an \(x\). So let's factor that out. 
\( 3( x \cdot x + 2 \cdot x) = 3x(x + 2) \) 
Now, let's again look at what is inside the parentheses. Notice that we have two terms with nothing in common. One has \(x\). The other has \(2\). Since they do not have anything in common, we are done. 
Okay, so that is how we would factor this polynomial. Here is a video showing the GCF method. See which why you think is easier to understand. However, as usual, check with your instructor to see what they require.
video by Freshmen Math Doctor 

Final Answer 

\( 3x^2 + 6x = 3x(x + 2) \)
close solution

Log in to rate this practice problem and to see it's current rating. 

Before we get into the details of factoring, let's watch this video clip as an overview. This instructor shows several techniques with a few quick examples. If you don't understand everything in this video, that's okay. We cover most of this techniques on this page and give you a chance to practice them.
video by freeCodeCamp.org 

Factoring By GCF (Greatest Common Factor)
Before we go on, we want to mention this technique. We have recently watched videos with instructors using this technique (including the last video above). Many instructors just seem to wave their hands and come up with the GCF of each term. We recommend instead to use the stepbystep, one factor at a time technique we demonstrated in the previous example. Essentially, it is the same as GCF but you do not need to come up with the one large factor all at once. Of course, make sure you check with your instructor to see what they expect.
Testing Linear Factors
Before we get into factoring techniques, there is one concept that will help you a lot. Let's say you have a polynomial and you suspect that \((x1)\) is a factor. What you can do is set this equal to zero and solve for \(x\), i.e. \( x1=0 \to x=1\) and substitute \(x=1\) into the polynomial. If the result is zero, then \((x1)\) is a factor of the polynomial. Use long division of polynomials or synthetic division to factor it out. This will reduce the highest power by one and perhaps give you a polynomial that you can then factor using simpler techniques.
So your next question is, why would I think that \((x1)\) might be a factor? Well, one idea is to plot the polynomial on your calculator (if your instructor allows it) and see where it might cross the xaxis, i.e. try to see if you can find any real zeroes/roots. This can reduce the complexity of the polynomial until it is more manageable.
Factoring Quadratics/Trinomials (Primary)
You will see quadratics often in calculus. There are several techniques that you can use on quadratics. Check out these practice problems for examples.
Unless otherwise instructed, factor these polynomials. Give your answers in simplified, completely factored form.
\( x^2  4x  5 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^2  4x  5 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^2 + 8x + 15 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^2 + 8x + 15 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^2  14x + 45 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^2  14x + 45 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

Solve \( x^2 = 11x  10 \) by factoring.
Problem Statement 

Solve \( x^2 = 11x  10 \) by factoring.
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

Solve \( x^2  13x + 36 = 0 \) by factoring.
Problem Statement 

Solve \( x^2  13x + 36 = 0 \) by factoring.
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 6x^4  18x^3 + 12x^2 \)
Problem Statement 

Unless otherwise instructed, factor \( 6x^4  18x^3 + 12x^2 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^2+4x  12 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^2+4x  12 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 3x^2 + 12x  36 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 3x^2 + 12x  36 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 3x^2 + 10x  8 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 3x^2 + 10x  8 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 8x^2 + 35x + 12 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 8x^2 + 35x + 12 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 6x^2  3x  45 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 6x^2  3x  45 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

Factor By Grouping (Secondary)
When it appears that you have groups of similar terms, try pairing them up and factoring them together and then seeing if you have the same terms in each group. This technique is best seen by example. Look at the first couple of practice problems for examples.
Unless otherwise instructed, factor these polynomials. Give your answers in simplified, completely factored form.
\( x^3  x^2  5x + 5 \)
Problem Statement 

Factor \( x^3  x^2  5x + 5 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^3  3x^2 + 4x  12 \)
Problem Statement 

Factor \( x^3  3x^2 + 4x  12 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^3 + 2x^2  5x  10 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^3 + 2x^2  5x  10 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 4x^3  8x^2 + 6x  12 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 4x^3  8x^2 + 6x  12 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^3 + 3x^2  4x  12 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^3 + 3x^2  4x  12 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^3  4x^2 + x + 6 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^3  4x^2 + x + 6 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 5v^3  2v^2 + 25v  10 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 5v^3  2v^2 + 25v  10 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 5r^4  7r^2s  6s^2 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 5r^4  7r^2s  6s^2 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 6xy  9y  10x 15 \)
Problem Statement 

Unless otherwise instructed, factor \( 6xy  9y  10x 15 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 15  5A^2  3B^2 + A^2B^2 \)
Problem Statement 

Unless otherwise instructed, factor \( 15  5A^2  3B^2 + A^2B^2 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

Difference of Two Squares (Secondary)
A special case that you will definitely come across in calculus is the difference of two squares.
The difference of two squares equation is pretty easy.
\( a^2  b^2 = (a+b)(ab) \) 

Note  This will not work on the sum of two squares.
See these practice problems for examples.
Unless otherwise instructed, factor these polynomials. Give your answers in simplified, completely factored form.
\( x^4  81 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^4  81 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 36x^2  49y^2 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 36x^2  49y^2 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 64x^2  81 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 64x^2  81 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^2  144 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( x^2  144 \)
Solution 

video by Freshmen Math Doctor 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 16x^2  81y^4 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 16x^2  81y^4 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 9x^2  49 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 9x^2  49 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 36y^4  100 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 36y^4  100 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 16x^2  25y^2 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 16x^2  25y^2 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 64x^2  81y^2 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 64x^2  81y^2 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( (x3)^2  4 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( (x3)^2  4 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( (x+7)^2  25 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( (x+7)^2  25 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( (x+2)^2  25 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( (x+2)^2  25 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

Sum And Difference of Cubes (Secondary)
Factoring two terms, both of them cubes does not come up very often in calculus. But you will see it. We include this section to give you of the equations and you can come back here when you need to remind yourself how to factor these types of equations.
\( a^3 + b^3 = (a + b)(a^2  ab + b^2) \) 

\( a^3  b^3 = (a  b)(a^2 + ab + b^2) \) 
For a discussion and some good examples of the sum and difference of cubes, go to this Purple Math page .
Unless otherwise instructed, factor these polynomials. Give your answers in simplified, completely factored form.
\( x^3  27 \)
Problem Statement 

Unless otherwise instructed, factor \( x^3  27 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 8x^3  64 \)
Problem Statement 

Unless otherwise instructed, factor \( 8x^3  64 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 8t^3 \)
Problem Statement 

Unless otherwise instructed, factor \( 8t^3 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 125A^3 + 27B^3 \)
Problem Statement 

Unless otherwise instructed, factor \( 125A^3 + 27B^3 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 64x^3 + 125 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 64x^3 + 125 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 27y^3  8 \)
Problem Statement 

Unless otherwise instructed, factor \( 27y^3  8 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 8x^3 + 27 \)
Problem Statement 

Unless otherwise instructed, factor \( 8x^3 + 27 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 27x^3 + 64y^3 \)
Problem Statement 

Unless otherwise instructed, factor \( 27x^3 + 64y^3 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 64y^3  125 \)
Problem Statement 

Unless otherwise instructed, factor \( 64y^3  125 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( 8y^327 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( 8y^327 \)
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

\( x^6  64y^9 \)
Problem Statement 

Unless otherwise instructed, factor \( x^6  64y^9 \)
Final Answer 

\( (x^24y^3)(x^4+4x^2y^3+16y^6) \)
Problem Statement 

Unless otherwise instructed, factor \( x^6  64y^9 \)
Solution 

Final Answer 

\( (x^24y^3)(x^4+4x^2y^3+16y^6) \)
close solution

Log in to rate this practice problem and to see it's current rating. 

\( (2x+3y)^3  343 \)
Problem Statement 

Unless otherwise instructed, factor the polynomial \( (2x+3y)^3  343 \)
Solution 

Not all instructors require students to multiply out the larger term. Check with your instructor to see what they expect.
close solution

Log in to rate this practice problem and to see it's current rating. 

Really UNDERSTAND Precalculus
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. 
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes 

Practice Instructions
Unless otherwise instructed, factor these polynomials. Give your answers in simplified, completely factored form.