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) (a
^{2} + b
^{2} - 2ab),
[Using the formula of (a + b)
^{2} = a
^{2} - 2ab + b
^{2}]
= a (a
^{2} + b
^{2} – 2ab) – b (a
^{2} + b
^{2} – 2ab)
= a
^{3} + ab
^{2} – 2a
^{2}b – ba
^{2} – b
^{3}
+ 2ab
^{2}
= a
^{3} – 3a
^{2}b + 3ab
^{2} – b
^{3}
Therefore, (a - b)
^{3} = a
^{3} – 3a
^{2}b + 3ab
^{2} – b
^{3}
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} = a
^{3} – 3a
^{2}b + 3ab
^{2} – b
^{3}
= a
^{3} – b
^{3} – 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} = a
^{3} – 3a
^{2}b + 3ab
^{2} – b
^{3}
(3x – 4y)
^{3}
Here, a = 3x, b = 4y
= (3x)
^{3} – 3 (3x)
^{2} (4y) + 3 (3x) (4y)
^{2} – (4y)
^{3}
= 27x
^{3} – 3 (9x
^{2}) (4y) + 3 (3x) (16y
^{2}) – 64y
^{3}
= 27x
^{3} – 108x
^{2}y + 144xy
^{2} – 64y
^{3}
Therefore, (3x – 4y)
^{3} = 27x
^{3} – 108x
^{2}y + 144xy
^{2} – 64y
^{3}
2. Use the formula and evaluate (997)
^{3}
Solution:
(997)
^{3} = (1000 – 3)
^{3}
We know, (a - b)
^{3} = a
^{3} – 3a
^{2}b + 3ab
^{2} – b
^{3}
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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