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

3x^{2} + 6x = 3x(x + 2) 
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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. 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 a large factor all at once. Of couse, 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 let \(x=1\), which makes \((x1)\) equal zero, 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.
Unless otherwise instructed, factor the polynomial \( x^2  4x  5 \).
Problem Statement 

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

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

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

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

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

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Unless otherwise instructed, solve \( x^2 = 11x  10 \) by factoring.
Problem Statement 

Unless otherwise instructed, solve \( x^2 = 11x  10 \) by factoring.
Solution 

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Unless otherwise instructed, solve \( x^2  13x + 36 = 0 \) by factoring.
Problem Statement 

Unless otherwise instructed, solve \( x^2  13x + 36 = 0 \) by factoring.
Solution 

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

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

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

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

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

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

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

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

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

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

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

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

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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 by grouping.
Factor \( x^3  x^2  5x + 5 \)
Problem Statement 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.
Unless otherwise instructed, factor the polynomial \( x^4  81 \).
Problem Statement 

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

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

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

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Unless otherwise instructed, factor the polynomial \( 64x^2  81 \).
Problem Statement 

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

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Unless otherwise instructed, factor the polynomial \( x^2  144 \).
Problem Statement 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 good examples of the sum and difference of cubes, go to this Purple Math page .
Unless otherwise instructed, factor these polynomials.
Unless otherwise instructed, factor \( x^3  27 \)
Problem Statement 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\( (x^24y^3)(x^4+4x^2y^3+16y^6) \) 
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Unless otherwise instructed, factor the polynomial \( (2x+3y)^3  343 \)
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.
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