%5E3-binomial-expansion.jpeg "(x-y)^3 Binomial Expansion")
Binomial Expansion of (x-y)^3
In algebra, the binomial expansion of (x-y)^3 is a fundamental concept that plays a crucial role in various mathematical operations. In this article, we will delve into the world of binomial expansion and explore the formula, notation, and example of (x-y)^3.
What is Binomial Expansion?
Binomial expansion is a mathematical operation that involves expanding an expression of the form (a+b)^n, where a and b are variables, and n is a positive integer. The result of the expansion is a sum of terms, each consisting of a combination of a and b raised to non-negative integer powers.
Formula for Binomial Expansion
The formula for binomial expansion is given by:
(a+b)^n = a^n + na^(n-1)b + n(n-1)/2! a^(n-2)b^2 + ... + b^n
where n is a positive integer.
Binomial Expansion of (x-y)^3
Using the formula above, we can expand (x-y)^3 as follows:
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
Notation and Explanation
In the expansion above, we can see that:
x^3is the first term, which is obtained by raisingxto the power of 3.-3x^2yis the second term, which is obtained by multiplyingxraised to the power of 2 by-3andy.3xy^2is the third term, which is obtained by multiplyingxby3andyraised to the power of 2.-y^3is the fourth term, which is obtained by raisingyto the power of 3 and multiplying it by-1.
Example
Find the binomial expansion of (2x-3y)^3.
Solution
Using the formula above, we can expand (2x-3y)^3 as follows:
(2x-3y)^3 = (2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3
= 8x^3 - 72x^2y + 216xy^2 - 27y^3
Therefore, the binomial expansion of (2x-3y)^3 is 8x^3 - 72x^2y + 216xy^2 - 27y^3.
Conclusion
In conclusion, the binomial expansion of (x-y)^3 is a crucial concept in algebra that involves expanding an expression of the form (x-y)^3 using the formula for binomial expansion. By understanding the formula and notation, we can easily expand expressions of this form and apply it to various mathematical problems.
