%5E5-binomial-expansion.jpeg "(2x-3)^5 Binomial Expansion")
Binomial Expansion of (2x-3)^5
In mathematics, the binomial theorem is a formula for expanding powers of a binomial, which is an expression consisting of two terms. In this article, we will explore the binomial expansion of (2x-3)^5.
What is Binomial Expansion?
Binomial expansion is a way of expanding an expression of the form (a+b)^n, where a and b are constants or variables, and n is a positive integer. The expansion involves raising each term in the binomial to the power of n and multiplying them together.
The Binomial Theorem
The binomial theorem is a formula that allows us to expand (a+b)^n in a more efficient way. It is given by:
(a+b)^n = a^n + na^(n-1)b + (n(n-1)/2!)a^(n-2)b^2 + … + b^n
This formula can be used to expand any binomial raised to a positive integer power.
Binomial Expansion of (2x-3)^5
Now, let's apply the binomial theorem to expand (2x-3)^5.
(2x-3)^5 = (2x)^5 + 5(2x)^4(-3) + (5(4)/2!)(2x)^3(-3)^2 + (5(4)(3)/3!)(2x)^2(-3)^3 + (5(4)(3)(2)/4!)(2x)(-3)^4 + (-3)^5
Expanding the Terms
Let's expand each term in the expression above:
(2x)^5 = 32x^5
5(2x)^4(-3) = 5(16x^4)(-3) = -240x^4
(5(4)/2!)(2x)^3(-3)^2 = (10)(8x^3)(9) = 720x^3
(5(4)(3)/3!)(2x)^2(-3)^3 = (10)(6)(4x^2)(-27) = -648x^2
(5(4)(3)(2)/4!)(2x)(-3)^4 = (10)(3)(2)(2x)(81) = 972x
(-3)^5 = -243
Final Expansion
Now, let's combine the expanded terms:
(2x-3)^5 = 32x^5 - 240x^4 + 720x^3 - 648x^2 + 972x - 243
This is the binomial expansion of (2x-3)^5.
Conclusion
In this article, we learned how to apply the binomial theorem to expand (2x-3)^5. We saw how the expansion involves raising each term in the binomial to the power of 5 and multiplying them together. The final expansion is a polynomial expression with six terms.
