Math Doubts

Factoring by the sum of cubes

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

Introduction

In mathematics, the sum of two cubes are appeared as a polynomial and it has to simplify in some cases. Mathematically, it is possible to express the sum of two cubes as a product of two expressions by factorization.

Required knowledge

For factorizing (or factorising) the sum of two cubes, you have to learn the following mathematical concept.

  1. Sum of cubes formula

In mathematics, the sum of two cubes formula is written in algebraic form in two ways.

  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

The factorisation (or factorization) of an expression that contains two cubes can be performed in two simple steps.

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

Example

Factorize $64x^3+1$

Step – 1

The given polynomial $64x^3+1$ has two terms but they both are not completely in the form of sum of two cubes but they can be expressed in sum of two cubes form by exponentiation.

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

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

Step – 2

Now, use the sum of two cubes formula to factorise it. According to $a^3+b^3$ $\,=\,$ $(a+b)(a^2+b^2-ab)$.

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

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

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

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

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