## 17Calculus Precalculus - Factoring Polynomials

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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 $$3x^2 + 6x$$

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

3x2 + 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 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.

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

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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

### 2544 video

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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

### 2540 video

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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

### 2536 video

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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

### 2537 video

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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

### 2541 video

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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

### 2653 video

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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

### 2654 video

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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

### 2655 video

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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

### 2662 video

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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

### 2663 video

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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

### 2545 video

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Factor $$x^3 - 3x^2 + 4x - 12$$

Problem Statement

Factor $$x^3 - 3x^2 + 4x - 12$$

Solution

### 2546 video

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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

### 2656 video

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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

### 2657 video

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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

### 2658 video

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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

### 2659 video

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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

### 2660 video

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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

### 2661 video

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Unless otherwise instructed, factor $$6xy - 9y - 10x -15$$

Problem Statement

Unless otherwise instructed, factor $$6xy - 9y - 10x -15$$

Solution

### 2664 video

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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

### 2665 video

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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.

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

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

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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

### 2548 video

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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

### 2549 video

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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

### 2550 video

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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

### 2645 video

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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

### 2646 video

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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

### 2647 video

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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

### 2648 video

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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

### 2649 video

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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

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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

### 2651 video

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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

### 2652 video

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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

### 2666 video

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Unless otherwise instructed, factor $$8x^3 - 64$$

Problem Statement

Unless otherwise instructed, factor $$8x^3 - 64$$

Solution

### 2667 video

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Unless otherwise instructed, factor $$8-t^3$$

Problem Statement

Unless otherwise instructed, factor $$8-t^3$$

Solution

### 2668 video

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Unless otherwise instructed, factor $$125A^3 + 27B^3$$

Problem Statement

Unless otherwise instructed, factor $$125A^3 + 27B^3$$

Solution

### 2669 video

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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

### 2670 video

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Unless otherwise instructed, factor $$27y^3 - 8$$

Problem Statement

Unless otherwise instructed, factor $$27y^3 - 8$$

Solution

### 2671 video

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Unless otherwise instructed, factor $$8x^3 + 27$$

Problem Statement

Unless otherwise instructed, factor $$8x^3 + 27$$

Solution

### 2672 video

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Unless otherwise instructed, factor $$27x^3 + 64y^3$$

Problem Statement

Unless otherwise instructed, factor $$27x^3 + 64y^3$$

Solution

### 2673 video

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Unless otherwise instructed, factor $$64y^3 - 125$$

Problem Statement

Unless otherwise instructed, factor $$64y^3 - 125$$

Solution

### 2674 video

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Unless otherwise instructed, factor the polynomial $$8y^3-27$$

Problem Statement

Unless otherwise instructed, factor the polynomial $$8y^3-27$$

Solution

### 2675 video

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Unless otherwise instructed, factor $$x^6 - 64y^9$$

Problem Statement

Unless otherwise instructed, factor $$x^6 - 64y^9$$

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

Problem Statement

Unless otherwise instructed, factor $$x^6 - 64y^9$$

Solution

### 2676 video

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

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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.

### 2677 video

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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

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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