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Sum of Cubes Formula

Sum of cubes formula is given by computing the area of the region in two ways: by squaring the length of a side and by adding the areas of the smaller squares. In other words, the sum of the first n natural numbers is the sum of the first n cubes.

The other name of sum of cube is factoring formula. The find the sum of cubes of any polynomial the given formula is used:

\[\large a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right)\]

solved examples

Question: Factor $27x^{3}+1$

Solution:

$27x^{3}+1=\left(3x\right)^{3}+1^{3}$

$=\left(3x+1\right)\left(\left(3x\right)^{2}-\left(3x\right)\left(1\right)+1^{2}\right )$

$=\left(3x+1\right)\left(9x^{2}-3x+1\right )$