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.

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.

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.

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

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

Latest Math Topics

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.