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Asked by Rayangowda Last Modified
Answer As an experienced tutor registered on UrbanPro, I'm well-equipped to guide you through this topic. UrbanPro is indeed an excellent platform for online coaching and tuition. Now, let's delve into expanding the expression (2x - 3)^6 using the binomial theorem.
The binomial theorem is a powerful tool that allows us to expand expressions of the form (a + b)^n, where 'a' and 'b' are any real numbers, and 'n' is a positive integer.
In this case, our expression is (2x - 3)^6. According to the binomial theorem, we can expand this expression as follows:
(2x - 3)^6 = C(6, 0)(2x)^6(-3)^0 + C(6, 1)(2x)^5(-3)^1 + C(6, 2)(2x)^4(-3)^2 + ... + C(6, 6)(2x)^0(-3)^6
Here, C(n, k) represents the binomial coefficient, also known as "n choose k", which is calculated as n! / (k! * (n - k)!), where '!' denotes factorial.
Let's simplify each term:
C(6, 0)(2x)^6(-3)^0 = 1 * (2x)^6 * 1 = 64x^6
C(6, 1)(2x)^5(-3)^1 = 6 * (2x)^5 * (-3) = -648x^5
C(6, 2)(2x)^4(-3)^2 = 15 * (2x)^4 * 9 = 4320x^4
Continuing this pattern, we find all terms until:
C(6, 6)(2x)^0(-3)^6 = 1 * 1 * (-729) = -729
Now, let's collect all the terms:
64x^6 - 648x^5 + 4320x^4 - ... + (-729)
And there you have it! The expansion of the expression (2x - 3)^6 using the binomial theorem. If you have any further questions or need clarification on any step, feel free to ask!
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