Sum of Cubes Formula and Solved Examples - Testbook

Last Updated on Jul 31, 2023
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The sum of cubes formula is a mathematical concept that calculates the total of the cubes of the first n natural numbers. It is based on the principle of calculating the volume of a region in two ways: by cubing the length of a side and by adding the volumes of the smaller cubes.

A Closer Look at the Sum of Cubes Formula

Also known as the factoring formula, the sum of cubes formula is a tool used to find the sum of cubes of any polynomial. Here is the formula for reference:

a 3 + b 3 = (a + b) (a 2 − ab + b 2 )

Examples to Understand the Sum of Cubes Formula

Example 1: Factor 8 x 3 + 64

Solution:

8 x 3 + 64 = (2 x) 3 + 4 3

= (2 x+4)[(2 x) 2 − (2 x)(4) + 4 2 )] = (2 x + 4)(4 x 2 – 8 x + 16)

Example 2: Factor: 125x 3 +1000

Solution:
125x 3 + 1000 = (5x) 3 + 10 3
Thus,
125x 3 + 1000 = (5x + 10)[(5x) 2 – (5x)(10) + 10 2 ] = (5x + 10)(25x 2 – 50x + 100)

Example 3: Factor: 27x 3 + 64y 3

Solution:
27x 3 + 64y 3 = (3x) 3 + (4y) 3
Thus,
27x 3 + 64y 3 = (3x + 4y)[(3x) 2 – (3x)(4y) + (4y) 2 ] = (3x + 4y)(9x 2 – 12xy + 16y 2 )

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Frequently Asked Questions

The formula for the sum of cubes is a^3 + b^3 = (a + b) (a^2 − ab + b^2).

The sum of cubes formula is used in factorization by equating the given polynomial to the formula, identifying the values of 'a' and 'b', and then simplifying the terms.

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