%5E5.jpeg "(2x+3)^5")
Expanding (2x+3)^5 using the Binomial Theorem
The Binomial Theorem provides a powerful tool for expanding expressions of the form (a + b)^n. In this case, we want to expand (2x + 3)^5. Let's break down the steps:
Understanding the Binomial Theorem
The Binomial Theorem states:
(a + b)^n = ∑_(k=0)^n (n choose k) * a^(n-k) * b^k
where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This represents the number of ways to choose k items from a set of n items.
- ∑_(k=0)^n represents the sum from k = 0 to n.
Applying the Theorem to (2x + 3)^5
- Identify a and b: In our case, a = 2x and b = 3.
- Identify n: n = 5.
- Calculate the binomial coefficients: We need to calculate (5 choose k) for k = 0, 1, 2, 3, 4, and 5:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
- Apply the formula: Substitute the values into the Binomial Theorem formula:
(2x + 3)^5 = (1 * (2x)^5 * 3^0) + (5 * (2x)^4 * 3^1) + (10 * (2x)^3 * 3^2) + (10 * (2x)^2 * 3^3) + (5 * (2x)^1 * 3^4) + (1 * (2x)^0 * 3^5)
Simplifying the Expression
Now, we can simplify the terms by performing the calculations:
(2x + 3)^5 = 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243
Conclusion
Therefore, the expanded form of (2x + 3)^5 is 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243. The Binomial Theorem provides a systematic way to expand expressions like this, simplifying the process and reducing the chance of error.