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.
7th Grade Math Problems
8th Grade Math Practice
From Cube of The Difference of Two Binomials to HOME PAGE
Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.