%5E5-binomial-theorem.jpeg "(2x+3)^5 Binomial Theorem")
Expanding (2x+3)^5 Using the Binomial Theorem
The binomial theorem provides a powerful tool for expanding expressions of the form (x+y)^n. Let's explore how to apply it to the specific case of (2x+3)^5.
Understanding the Binomial Theorem
The binomial theorem states that for any real numbers x and y, and any non-negative integer n:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
where (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!). This coefficient represents the number of ways to choose k objects out of n.
Applying the Theorem to (2x+3)^5
To expand (2x+3)^5, we'll use the binomial theorem with x = 2x, y = 3, and n = 5. Let's break down the process:
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Identify the terms:
- x = 2x
- y = 3
- n = 5
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Calculate the binomial coefficients:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
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Apply the theorem:
(2x + 3)^5 = (5 choose 0) * (2x)^5 * 3^0 + (5 choose 1) * (2x)^4 * 3^1 + (5 choose 2) * (2x)^3 * 3^2 + (5 choose 3) * (2x)^2 * 3^3 + (5 choose 4) * (2x)^1 * 3^4 + (5 choose 5) * (2x)^0 * 3^5
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Simplify:
(2x + 3)^5 = 1 * 32x^5 * 1 + 5 * 16x^4 * 3 + 10 * 8x^3 * 9 + 10 * 4x^2 * 27 + 5 * 2x * 81 + 1 * 1 * 243
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Final expansion:
(2x + 3)^5 = 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243
Conclusion
By applying the binomial theorem, we have successfully expanded (2x+3)^5. The process involved identifying the terms, calculating the binomial coefficients, and then substituting the values into the theorem formula. The final result is a polynomial expression with each term representing a specific combination of the powers of 2x and 3.