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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.

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