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Grade 12

Factor sum and difference of cubes

Lesson

We've looked at how to factor monic and non-monic quadratics. We've also looked at some special factoring rules, such as the difference of two squares.

In this chapter we are going to look at how to factor the sum and difference of two cubics (ie. terms with powers of $3$3).

Let's start by looking at the general forms of these rules.

General Forms for Factoring Cubics

Sum of two cubics: $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$a3+b3=(a+b)(a2ab+b2)

Difference of two cubics: $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$a3b3=(ab)(a2+ab+b2)

Some people use the mnemonic "SOAP" to help remember the order of the signs in these formulae. The letters stand for:

SAME as the sign in the middle of the original expression

OPPOSITE sign to the original expression

ALWAYS POSITIVE

Now let's look at some examples and see this process in action!

 

Worked Examples

Question 1

Factor $x^3+27$x3+27.

Question 2

Factor $x^3-1000$x31000.

Question 3

Evaluate $5^3-11^3$53113 by factoring it first.

Note: You must show your factoring.

Question 4

Simplify $\frac{125x^3+8}{5x+2}$125x3+85x+2.

 

 

Outcomes

12F.C.3.2

Factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies

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