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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