%5E5-binomial-theorem.jpeg "(x+3)^5 Binomial Theorem")
Expanding (x + 3)^5 Using the Binomial Theorem
The binomial theorem is a powerful tool for expanding expressions of the form (x + y)^n. It allows us to quickly and efficiently find the coefficients of each term in the expansion.
Understanding the Binomial Theorem
The binomial theorem states that for any positive integer n:
(x + y)^n = ∑_(k=0)^n (n choose k) x^(n-k) y^k
Where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This is also often represented as "nCk".
- ∑_(k=0)^n denotes the sum from k = 0 to k = n.
Applying the Binomial Theorem to (x + 3)^5
Let's apply the binomial theorem to expand (x + 3)^5:
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Identify n and y: In this case, n = 5 and y = 3.
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Calculate binomial coefficients: We need to calculate (5 choose k) for k = 0, 1, 2, 3, 4, and 5. Here's how they look:
- (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 formula: Now, we can expand (x + 3)^5 by substituting the values:
(x + 3)^5 = (5 choose 0) x^5 (3)^0 + (5 choose 1) x^4 (3)^1 + (5 choose 2) x^3 (3)^2 + (5 choose 3) x^2 (3)^3 + (5 choose 4) x^1 (3)^4 + (5 choose 5) x^0 (3)^5
- Simplify:
(x + 3)^5 = 1x^5 + 5x^4(3) + 10x^3(9) + 10x^2(27) + 5x(81) + 1(243)
(x + 3)^5 = x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243
Therefore, the expanded form of (x + 3)^5 is x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243.
Key Points
- The binomial theorem provides a systematic way to expand binomials raised to any power.
- The binomial coefficients determine the coefficients of each term in the expansion.
- The exponents of x and y in each term add up to n.
By understanding and applying the binomial theorem, you can efficiently expand complex expressions and gain deeper insights into their algebraic structure.