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.

Formula to Find Sum of Cubes

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

 a3 + b3 = (a + b) (a2 − ab + b2)

Solved Example Question

Question:Factor 27 x3 + 1
Solution:
27 x3 + 1 = (3 x)3 + 13
= (3 x+1)[(3 x)2 − (3 x)(1) + 12)]
= (3 x + 1)(9 x2 – 3 x + 1)
Question 2: Factor: 8x3 +125
Solution:
8x3 + 125 = (2x)3 + 53
Comparing with a3 + b3,
a = 2x, b = 5
a3 + b3 = (a + b)(a2 – ab + b2)
Thus,
8x3 + 125 = (2x + 5)[(2x)2 – (2x)(5) + 52] = (2x + 5)(4x2 – 10x + 25)
Question 3: Factor: 64x3 + 27y3
Solution:
64x3 + 27y3 = (4x)3 + (3y)3
Comparing with a3 + b3,
a = 4x, b = 3y
a3 + b3 = (a + b)(a2 – ab + b2)
Thus,
64x3 + 27y3 = (4x + 3y)[(4x)2 – (4x)(3y) + (3y)2] = (4x + 3y)(16x2 – 12xy + 9y2)