\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Factoring Polynomials

Algebra

Polynomials

Functions

Rational Functions

Graphing

Matrices

Systems

Trigonometry

Complex Numbers

Applications

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

Algebra

Functions

Functions

Polynomials

Rational Functions

Graphing

Matrices & Systems

Matrices

Systems

Trigonometry & Complex Numbers

Trigonometry

Complex Numbers

Applications

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

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

3x2 + 6x = 3x(x + 2)

Problem Statement

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

Solution

First, we will break each factor down into individual terms.

3x2 = 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.

2542 video

Final Answer

3x2 + 6x = 3x(x + 2)

close solution

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

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 step-by-step, 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 \((x-1)\) is a factor. What you can do is let \(x=1\), which makes \((x-1)\) equal zero, and substitute \(x=1\) into the polynomial. If the result is zero, then \((x-1)\) 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 \((x-1)\) 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 x-axis, 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

2543 video

close solution

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

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

Problem Statement

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

Solution

2544 video

close solution

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

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

Problem Statement

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

Solution

2540 video

close solution

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

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

Problem Statement

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

Solution

2536 video

close solution

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

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

2537 video

close solution

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

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

Problem Statement

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

Solution

2541 video

close solution

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

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

Problem Statement

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

Solution

2653 video

close solution

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

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

Problem Statement

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

Solution

2654 video

close solution

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

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

Problem Statement

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

Solution

2655 video

close solution

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

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

Problem Statement

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

Solution

2662 video

close solution

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

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

Problem Statement

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

Solution

2663 video

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 by grouping.

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

Problem Statement

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

Solution

2545 video

close solution

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

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

Problem Statement

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

Solution

2546 video

close solution

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

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

2656 video

close solution

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

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

2657 video

close solution

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

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

2658 video

close solution

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

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

2659 video

close solution

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

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

2660 video

close solution

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

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

2661 video

close solution

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

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

Problem Statement

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

Solution

2664 video

close solution

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

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

2665 video

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.

\( a^2 - b^2 = (a+b)(a-b) \)

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

2547 video

close solution

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

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

Problem Statement

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

Solution

2548 video

close solution

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

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

Problem Statement

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

Solution

2549 video

close solution

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

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

Problem Statement

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

Solution

2550 video

close solution

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

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

Problem Statement

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

Solution

2645 video

close solution

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

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

Problem Statement

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

Solution

2646 video

close solution

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

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

Problem Statement

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

Solution

2647 video

close solution

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

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

Problem Statement

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

Solution

2648 video

close solution

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

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

Problem Statement

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

Solution

2649 video

close solution

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

Unless otherwise instructed, factor the polynomial \( (x-3)^2 - 4 \)

Problem Statement

Unless otherwise instructed, factor the polynomial \( (x-3)^2 - 4 \)

Solution

2650 video

close solution

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

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

Problem Statement

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

Solution

2651 video

close solution

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

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

Problem Statement

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

Solution

2652 video

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 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

2666 video

close solution

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

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

Problem Statement

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

Solution

2667 video

close solution

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

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

Problem Statement

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

Solution

2668 video

close solution

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

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

Problem Statement

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

Solution

2669 video

close solution

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

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

Problem Statement

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

Solution

2670 video

close solution

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

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

Problem Statement

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

Solution

2671 video

close solution

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

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

Problem Statement

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

Solution

2672 video

close solution

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

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

Problem Statement

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

Solution

2673 video

close solution

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

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

Problem Statement

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

Solution

2674 video

close solution

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

Unless otherwise instructed, factor the polynomial \( 8y^3-27 \)

Problem Statement

Unless otherwise instructed, factor the polynomial \( 8y^3-27 \)

Solution

2675 video

close solution

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

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

Problem Statement

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

Final Answer

\( (x^2-4y^3)(x^4+4x^2y^3+16y^6) \)

Problem Statement

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

Solution

2676 video

Final Answer

\( (x^2-4y^3)(x^4+4x^2y^3+16y^6) \)

close solution

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

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.

2677 video

close solution

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

Really UNDERSTAND Precalculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

To bookmark this page and practice problems, log in to your account or set up a free account.

Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Search Practice Problems

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.

math and science learning techniques

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.

The Practicing Mind: Developing Focus and Discipline in Your Life - Master Any Skill or Challenge by Learning to Love the Process

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Shop Amazon - Used Textbooks - Save up to 90%

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Try Amazon Music Unlimited Free Trial

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.