%5E4-binomial-expansion.jpeg "(2x-3y)^4 Binomial Expansion")
Binomial Expansion of (2x-3y)^4
In this article, we will explore the binomial expansion of (2x-3y)^4. Binomial expansion is a fundamental concept in algebra, and it is used to expand an expression of the form (a+b)^n, where a and b are variables and n is a positive integer.
Binomial Theorem
The binomial theorem states that:
(a+b)^n = a^n + na^(n-1)b + (n-1 choose 2)a^(n-2)b^2 + ... + b^n
where n choose k is the binomial coefficient, which can be calculated as:
n choose k = n! / (k!(n-k)!)
Expansion of (2x-3y)^4
Using the binomial theorem, we can expand (2x-3y)^4 as follows:
(2x-3y)^4 = (2x)^4 - 4(2x)^3(3y) + 6(2x)^2(3y)^2 - 4(2x)(3y)^3 + (3y)^4
Simplifying the expression, we get:
(2x-3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4
Step-by-Step Calculation
Here is the step-by-step calculation of the binomial expansion of (2x-3y)^4:
Term 1: a^4
(2x)^4 = 16x^4
Term 2: 4a^3b
4(2x)^3(3y) = 4(8x^3)(3y) = 96x^3y
Term 3: 6a^2b^2
6(2x)^2(3y)^2 = 6(4x^2)(9y^2) = 216x^2y^2
Term 4: 4ab^3
4(2x)(3y)^3 = 4(2x)(27y^3) = 216xy^3
Term 5: b^4
(3y)^4 = 81y^4
Combining the Terms
16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4
Conclusion
In this article, we have successfully expanded (2x-3y)^4 using the binomial theorem. The final result is:
(2x-3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4
This expansion can be useful in various algebraic manipulations and applications.
