Difference Of Two Cubes [Math, Algebra]

Hi there. In this math post I quickly cover the concept of difference of cubes. This is one of those niche math/algebra topics that comes up when it comes to simplifying algebra expressions.

This topic is more for high school students in Grade 11 or 12 and for university students in engineering, math and maybe science students.

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The Formula

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For the difference of cubes you need to have an expression of the form:

then you can factor this difference of cubes into:

 

Variation - Sum of Cubes

A variation of the difference of cubes is factoring sum of cubes.

 

The ab term sign changes in this variation. It is -ab.

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Examples

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Example One

Factor the difference of cubes

In this case x = a and b = 2 from b-cubed equal to 8. Apply the formula to get the factors.

 

Example Two

Factor .

Here we have x = a and y^2 = b. Apply the formula accordingly.

 

Example Three

Simplify .

The top part of the fraction is a difference of cubes. Factoring the top may reveal something helpful for simplifying the fraction.

On the top you have the cube root of x-cubed which is x. The cube root of 27 is 3. Apply the difference of cubes formula for the top.

 

You now have (x - 3) factors that can be cancelled on the top and bottom.

 

The fraction simplifies into a quadratic expression.

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Thank you for reading.