$a^3-b^3$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

Let $a$ and $b$ be two quantities in algebraic form.

- The subtraction of $b$ from the $a$ is written as $a-b$ in mathematical form.
- The subtraction of $b$ cube from $a$ cube is mathematically written as $a^3-b^3$.
- Add the sum of the squares of $a$ and $b$ to the product of $a$ and $b$, and their sum is written as $a^2+ab+b^2$.

As per the difference of cubes arithmetic property, the subtraction of $b$ cubed from $a$ cubed is equal to the product of the subtraction of $b$ from $a$ and the addition of sum of squares of $a$ and $b$ and the product of $a$ and $b$.

$\therefore\,\,\,$ $a^3-b^3$ $\,=\,$ $(a-b)$ $\times$ $(a^2+b^2+ab)$

Now, it is time to learn more about the $a$ cube minus $b$ cube algebraic identity.

The $a$ cube minus $b$ cube algebraic identity is alternatively written in mathematics as follows.

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

The $a$ cube minus $b$ cube algebraic identity is used in two different cases mainly.

- In basic mathematics, it is used as a formula to evaluate the difference of cubes of two quantities.
- It is also used as a formula to write the difference of two quantities in cube form as a product of two factors.

The $a$ cube minus $b$ cube algebraic identity can be derived in two different methods.

Learn how to derive the $a$ cube minus $b$ cube formula by the algebraic identities.

Learn how to prove the $a$ cubed minus $b$ cubed identity geometrically by the volume of a cube.

Assume that $a = 5$ and $b = 2$

$(1).\,\,$ $a-b$ $\,=\,$ $5-2$ $\,=\,$ $3$

$(2).\,\,$ $a^2+b^2+ab$ $\,=\,$ $5^2+2^2+5 \times 2$ $\,=\,$ $25+4+10$ $\,=\,$ $39$

$(3).\,\,$ $a^3-b^3$ $\,=\,$ $5^3-2^3$ $\,=\,$ $125-8$ $\,=\,$ $117$

Now, calculate the product of the $a-b$ and $a^2+b^2+ab$.

$\implies$ $(a-b)$ $\times$ $(a^2+b^2+ab)$ $\,=\,$ $3 \times 39$ $\,=\,$ $117$

It is calculated that the product of them is equal to $117$, and the difference of cubes of $a$ and $b$ is also equal to $117$.

$\,\,\,\therefore\,\,\,\,\,\,$ $a^3-b^3$ $\,=\,$ $117$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

You too can verify the $a$ cube minus $b$ cube algebraic identity by taking any two numbers as explained above.

Latest Math Topics

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved