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Worked-out examples for the expansion of cube of a binomial:Simplify the following by cubing:1. (x + 5y)3 + (x – 5y)3

Solution:

We know, (a + b)3 = a3 + 3a2b + 3ab2 + b3

and,

(a – b)3 = a3 – 3a2b + 3ab2 – b3

Here, a = x and b = 5y

Now using the formulas for cube of two binomials we get,

= x3 + 3.x2.5y + 3.x.(5y)2 + (5y)3 + x3 - 3.x2.5y + 3.x.(5y)2 - (5y)3

= x3 + 15x2y + 75xy2 + 125 y3 + x3 - 15x2y + 75xy2 - 125 y3

= 2x3 + 150xy2

Therefore, (x + 5y)3 + (x – 5y)3 = 2x3 + 150xy2

Solution:

We know, (a + b)3 = a3 + 3a2b + 3ab2 + b3

and,

(a – b)3 = a3 – 3a2b + 3ab2 – b3

Here, a = x and b = 5y

Now using the formulas for cube of two binomials we get,

= x3 + 3.x2.5y + 3.x.(5y)2 + (5y)3 + x3 - 3.x2.5y + 3.x.(5y)2 - (5y)3

= x3 + 15x2y + 75xy2 + 125 y3 + x3 - 15x2y + 75xy2 - 125 y3

= 2x3 + 150xy2

Therefore, (x + 5y)3 + (x – 5y)3 = 2x3 + 150xy2