# Proof of ${(a+b)}^3$ formula in Algebraic Method

The a plus b whole cube formula can be derived in algebraic approach by multiplying three same sum basis binomials. Here $a$ and $b$ represents two terms in algebraic form.

### Product form of Binomials

Multiplying the binomial $a+b$ by itself three times is the mathematical meaning of cube of the binomial $a+b$. So, the $a$ plus $b$ whole cube can be expressed in product form by multiplying three same binomials.

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

Multiplying three same binomials is a special case in mathematics. Hence, the product of them is often called as the special product of binomials.

### Multiplying the Algebraic expressions

It is not possible to multiply all three same binomials at a time. So, multiply any two binomials firstly and then multiply remaining two factors for getting expansion of ${(a+b)}^3$ identity in algebraic approach.

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $\Big({(a+b)}$ $\times$ ${(a+b)}\Big)$

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $\Big(a \times (a+b)$ $+$ $b \times (a+b)\Big)$

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $(a \times a$ $+$ $a \times b$ $+$ $b \times a$ $+$ $b \times b)$

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $(a^2$ $+$ $ab$ $+$ $ba$ $+$ $b^2)$

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $(a^2$ $+$ $ab$ $+$ $ab$ $+$ $b^2)$

$\implies$ ${(a+b)}^3$ $\,=\,$ ${(a+b)}$ $\times$ $(a^2+2ab+b^2)$

Now, multiply the sum of two terms with the expansion of $a+b$ whole square by the multiplication of algebraic expressions.

$\implies$ ${(a+b)}^3$ $\,=\,$ $a \times (a^2+2ab+b^2)$ $+$ $b \times (a^2+2ab+b^2)$

$\implies$ ${(a+b)}^3$ $\,=\,$ $a \times a^2$ $+$ $a \times 2ab$ $+$ $a \times b^2$ $+$ $b \times a^2$ $+$ $b \times 2ab$ $+$ $b \times b^2$

$\implies$ ${(a+b)}^3$ $\,=\,$ $a^3$ $+$ $2a^2b$ $+$ $ab^2$ $+$ $ba^2$ $+$ $2ab^2$ $+$ $b^3$

$\implies$ ${(a+b)}^3$ $\,=\,$ $a^3$ $+$ $2a^2b$ $+$ $ab^2$ $+$ $a^2b$ $+$ $2ab^2$ $+$ $b^3$

$\implies$ ${(a+b)}^3$ $\,=\,$ $a^3$ $+$ $b^3$ $+$ $2a^2b$ $+$ $a^2b$ $+$ $ab^2$ $+$ $2ab^2$

$\,\,\, \therefore \,\,\,\,\,\,$ ${(a+b)}^3$ $\,=\,$ $a^3$ $+$ $b^3$ $+$ $3a^2b$ $+$ $3ab^2$

Thus, the $a+b$ whole cube formula is expanded in terms of $a$ and $b$ in mathematics.

### Simplified form of the Expansion

The expansion of cube of sum of terms $a$ and $b$ can also be written as the following simplified form.

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

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