Cube of Sum of Two terms

The sum of two terms raised to the power of $3$ is called the cube of sum of two terms.

Introduction

In mathematics, two terms are connected by a plus sign to form a binomial and the binomials are sometimes raised to the power of $3$ for representing the product of them in simple form. It is called cube of sum of two terms or also called as the sum of two terms whole cubed.

Example

$2p$ and $3q$ are two terms and their sum is $2p+3q$. It is a binomial and cube of this binomial is expressed in mathematics as follows.

${(2p+3q)}^3$

The cube of the binomial represents the product of three same binomials.

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ ${(2p+3q)}$ $\times$ ${(2p+3q)}$

The equation in algebraic form represents that the cube of a binomial can be calculated by multiplying three same binomials. It can be done as per the multiplication of algebraic expressions.

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $\Big[$ ${(2p+3q)}$ $\times$ ${(2p+3q)}$ $\Big]$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $\Big[$ $2p \times (2p+3q)$ $+$ $3q \times (2p+3q)$ $\Big]$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $\Big[$ $2p \times 2p$ $+$ $2p \times 3q$ $+$ $3q \times 2p$ $+$ $3q \times 3q$ $\Big]$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $\Big[$ ${(2p)}^2$ $+$ $6pq$ $+$ $6qp$ $+$ ${(3q)}^2$ $\Big]$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $\Big[$ $4p^2+6pq+6pq+9q^2$ $\Big]$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p+3q)}$ $\times$ $(4p^2+12pq+9q^2)$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ $2p \times (4p^2+12pq+9q^2)$ $+$ $3q \times (4p^2+12pq+9q^2)$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ $2p \times 4p^2$ $+$ $2p \times 12pq$ $+$ $2p \times 9q^2$ $+$ $3q \times 4p^2$ $+$ $3q \times 12pq$ $+$ $3q \times 9q^2$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ $8p^3$ $+$ $24p^2q$ $+$ $18pq^2$ $+$ $12qp^2$ $+$ $36pq^2$ $+$ $27q^3$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ $8p^3$ $+$ $24p^2q$ $+$ $18pq^2$ $+$ $12p^2q$ $+$ $36pq^2$ $+$ $27q^3$

$\implies$ ${(2p+3q)}^3$ $\,=\,$ $8p^3$ $+$ $27q^3$ $+$ $24p^2q$ $+$ $12p^2q$ $+$ $18pq^2$ $+$ $36pq^2$

$\,\,\, \therefore \,\,\,\,\,\,$ ${(2p+3q)}^3$ $\,=\,$ $8p^3$ $+$ $27q^3$ $+$ $36p^2q$ $+$ $54pq^2$

The right-hand side of the equation can be simply written in the following mathematical form.

$\implies$ ${(2p+3q)}^3$ $\,=\,$ ${(2p)}^3+{(3q)}^3+18pq(2p+3q)$

Thus, the cube of sum of two terms can be expanded in terms of them for calculating its value mathematically. It is a time taking process but those who are newly learning algebraic mathematics must follow this method for practice and the advanced learners can easily get it from an algebraic identity.

Formula

The expansion of cube of sum of two terms or binomial is represented in general form in the following two popular algebraic identities. Remember, they both are same and you can use any one of them to get the expansion of cube of a binomial quickly.

$(1) \,\,\,$ ${(a+b)}^3$ $\,=\,$ $a^3+b^3+3ab(a+b)$

$(2) \,\,\,$ ${(x+y)}^3$ $\,=\,$ $x^3+y^3+3xy(x+y)$

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