Sums and Differences of Cubes

(Video) Factoring Sums and Differences of of Cubes

by PatrickJMT

Patick JMT has a very good video explaining how to factor sums and differences of cubes.

(Video) Sums and Differences of Cubes

by PatrickJMT

Patrick JMT has another good video explaining sum and differences of cubes with examples.

(Video) Factoring Sum and Difference of Cubes

by mathman1024

Here's a video by mathman1024 showing you how to factor polynomials with the sum or difference of cubes. The formulas he uses are: $$a^3+b^3=(a+b)(a^2-ab+b^2)\\ a^3-b^3=(a-b)(a^2+ab+b^2)$$ The first polynomial he looks at is $x^3-8$. So the first step is to put it into the form of something cubed $\pm$ something else cubed. Since this problem has a minus we use minus. First put down the skeleton: $$(\ \ \ )^3-(\ \ \ )^3$$ Next we find what term cubed gives us $x^3$. Obviously the answer is $x$, so we put that in the first slot. $$(x)^3-(\ \ \ )^3$$ Then we find what cubed gives us $8$. The answer is $2$ because $2*2*2=8$. $$(x)^3-(2)^3$$ Then we factor into the form given by the equation above. In this problem $a=x$ and $b=2$. So we factor this polynomial into: $$(x-2)(x^2+2x+4)$$ Note that when using these formulas the quadratic term will not be able to factored any further. In other words, we can't factor $(x^2+2x+4)$ anymore.

The second example he goes through is $343x^3+64y^3$. See if you can solve this one on your own and then watch the video to see if your answer is correct.

The next lesson in our series of factoring is factoring sums and differences of cubes!

Background

You may remember back in Algebra 1 when you learned about the difference of squares. This concept is very similar to that one, so keep that in mind as you move forward! Cubic functions are those which have a power of $3$. The standard form is for a sum of squares is:

$$a^3+b^3$$

And the standard form for a difference of squares is:

$$a^3-b^3$$

You are essentially going to factor out a linear factor, which will leave you with a quadratic factor. When factoring a sum of cubes, it's helpful to think: "same, opposite, always positive". You're about to see why!

Factoring Sums of Cubes

The formula for factoring a sum of cubes is:

$$a^3+b^3=(a+b)(a^2-2ab+b^2)$$

Think back to "same, opposite, always positive". The sign in the linear term that we factored out is the same as the sum of cubes, so it's positive. The second sign is the opposite, so it's negative. The third sign is always positive!

Let's walk through an example.

Example: Factor $8x^3+27$ using the sum of cubes rule.

We can first recognize that $a=2x$ and $b=3$.

$$\begin{align} 8x^3+27\\ (2x+3)(4x^2-6x+9)\\ \end{align}$$

Factoring Difference of Cubes

The formula for factoring a difference of cubes is:

$$a^3-b^3=(a-b)(a^2+2ab+b^2)$$

Think back to "same, opposite, always positive". The sign in the linear term that we factored out is the same as the difference of cubes, so it's negative. The second sign is the opposite, so it's positive. The third sign is always positive!

Example: Factor $x^3-64$ using the difference of cubes rule.

We can first recognize that $a=x$ and $4$.

$$\begin{align} x^3-64 \\ (x-4)(x^2+4x+16)\\ \end{align}$$

Sums and Differences of Cubes

We've covered the Factor Theorem! Another pattern we may see when factoring polynomials is the sum and difference of cubes!

It might be hard to keep the signs straight, but we have a nifty trick to keep track of signs! Always remember SOAP.

$$\text {Same, Opposite, Always Positive}$$

The first sign, in the binomial will be the same sign. So, for a sum of squares, the first sign will be positive. For the difference of squares, the first sign will be negative.

The second sign will be the opposite, and the last sign will always be positive!