Cube of the Difference of Two Binomials

What is the formula for the cube of the difference of two binomials?

To determine cube of a number means multiplying a number with itself three times similarly, cube of a binomial means multiplying a binomial with itself three times.



(a - b) (a - b) (a - b) = (a - b)3

or, (a - b) (a - b) (a - b) = (a - b) (a - b)2

                                   = (a – b) (a2 + b2 - 2ab),
                                   [Using the formula of (a + b) 2 = a2 - 2ab + b2]

                                   = a (a2 + b2 – 2ab) – b (a2 + b2 – 2ab)

                                   = a3 + ab2 – 2a2b – ba2 – b3 + 2ab2

                                   = a3 – 3a2b + 3ab2 – b3

Therefore, (a - b)3 = a3 – 3a2b + 3ab2 – b3

Thus, we can write it as; a = first term, b = second term

(First term – Second term)3 = (first term)3 - 3 (first term)2 (second term) + 3 (first term) (second term)2 - (second term)3

So, the formula for the cube of the difference of two terms is written as:

(a - b)3 = a3 – 3a2b + 3ab2 – b3

            = a3 – b3 – 3ab (a - b)


Worked-out examples to find the cube of the difference of two binomials:

1. Determine the expansion of (3x – 4y)3

Solution:

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

(3x – 4y)3

Here, a = 3x, b = 4y

= (3x)3 – 3 (3x)2 (4y) + 3 (3x) (4y)2 – (4y)3

= 27x3 – 3 (9x2) (4y) + 3 (3x) (16y2) – 64y3

= 27x3 – 108x2y + 144xy2 – 64y3

Therefore, (3x – 4y)3 = 27x3 – 108x2y + 144xy2 – 64y3


2. Use the formula and evaluate (997)3

Solution:

(997)3 = (1000 – 3)3

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

Here, a = 1000, b = 3

(1000 – 3)3

= (1000)3 – 3 (1000)2 (3) + 3 (1000) (3)2 – (3)3

= 1000000000 – 9 (1000000) + (3000) 9 – 27

= 1000000000 – 9000000 + 27000 – 27

= 991026973

Therefore, (997)3 = 991026973

Thus, to expand the cube of the difference of two binomials we can use the formula to evaluate.

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