Factoring by the sum of cubes

A method of factoring the sum of two cubes as a product of two factors is called the factorization (or factorisation) by the sum of cubes.

Introduction

A polynomial can be formed by the sum of two cube quantities. Due to the involvement of the indeterminate variables, it is not possible to add one term in cube form to another term in cube form. However, the sum of two cubes is written as a product of two factors in mathematics and this process is called the factorisation (or factorization) by the sum of cubes.

Required knowledge

It is essential to have an idea about the factoring formula for sum of cubes to factorize (or factorise) the sum of two cubes. The sum of two cubes formula is written in mathematics in the following two forms.

1. $a^3+b^3$ $\,=\,$ $(a+b)(a^2+b^2-ab)$
2. $x^3+y^3$ $\,=\,$ $(x+y)(x^2+y^2-xy)$

Steps

Any expression in the form of sum of two cubes can be factored easily by following the below two simple steps.

1. Write each term of the expression in cube form.
2. Factorize the polynomial as a product of two expressions by using sum of two cubes rule.

Example

Let us understand how to factorize a polynomial by the sum of two cubes formula from the following example.

Factorize $64x^3+1$

Write every term in cube form

There is a variable in cube form in the first term of the polynomial. Hence, let’s try to express the expression by writing each term in cube form.

$= \,\,\,$ $4^3x^3+1^3$

$= \,\,\,$ $(4x)^3+1^3$

Factorize by the sum of cubes rule

It is time to factorise the sum of the two cubes by using the factoring formula for sum of two cubes.

Take $a \,=\, 4x$ and $b \,=\, 1$

Now, substitute them in the sum of two cubes formula.

$\implies$ $(4x)^3+1^3$ $\,=\,$ $(4x+1)\big((4x)^2+(1)^2-(4x)(1)\big)$

$\,\,\, \therefore \,\,\,\,\,\,$ $(4x)^3+1^3$ $\,=\,$ $(4x+1)(16x^2+1-4x)$

Therefore, the algebraic expression $(4x)^3+1^3$ is factored as $(4x+1)(16x^2+1-4x)$ mathematically by the sum of two cubes.

Problems

List of the questions on factorization by the sum of two cubes with solutions to learn how to express the sum of two cubes in factor form.

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