How to Factor x3 + 8: The Sum of 2 Cubes

In this article, we will discuss a method for factorizing x^3 + 8. This is the sum of 2 cubes, so it can be factorized using the sum of 2 cubes formula.

We will start by learning the formula and look at some examples and tips from a maths teacher with over 10 years of experience. Once we have done that, we can look at an alternate method of factoring x^3+8. Before we then learn the difference of 2 cubes formula. Let’s get started!

The sum of 2 cubes formula

The sum of 2 cubes formula is:

This is used to factorize the sum of 2 cubes.

In the first bracket is the sum of the cube root of each term.

In the second bracket, the first term is the square of the first term in the first bracket, the middle term is the product of the 2 terms from the first bracket, and the last term is the square of the last term in the first bracket.

Using SOAP to remember the signs

SOAP is an acronym that stands for

Same

Opposite

Always

Positive

You can use SOAP to remember what sign to use in each bracket.

Examples of factorizing using the sum of 2 cubes formula

Let’s look at a few examples.

1) Factorize x^3 + 1

2) Factorize x^3 +27

3) Factorize 2a^3 + 250

4) Factorize b^3 + 1000

Tips to know when to use the sum of 2 cubes

• It helps to know the first 5 or so cubic numbers so you can recognize them. Then when you see the sum of 2 cubic terms you can use this formula and SOAP to factorize it.

• Look for a common factor first, like in example 3 above.
• Use SOAP to remember when to use a positive sign and when to use a negative sign.

Alternate method: Using the factor theorem and equating coefficients

If you let f(x) = x^3+8

f(-2) = 0.

So by the factor theorem (x+2) is a factor of f(x).

Therefore x^3+8=(x+2)(ax^2+bx+c)

= ax^3+bx^2+cx+2ax^2+2bx+2c

=ax^3+(b+2a)x^2+(2b+c)x+2c

Equating coefficients,

a = 1

b+2a=0

2b+c=0

2c=8

Therefore b+2(1)=0, so b = -2

2b+c = 0, so 2(-2)+c=0, so c=4

Therfore x^3+8=(x+2)(x^2-2x+4)

The difference of two cubes formula

This is used to factorize the difference of 2 cubic terms.

It is similar to the sum of 2 cubes formula.

Just the signs are different so SOAP is useful to remember where to use a positive or negative sign.

Examples of factorizing using the difference of 2 cubes formula

Let’s look at a few examples where we can use this formula

1) Factorize x^3-1

2) Factorize a^3-27

FAQs

Further factors

Learn more about the factors of other numbers here.