%5E5-expand.jpeg "(x+1)^5 Expand")
Expanding (x+1)^5
The expression (x+1)^5 represents the product of (x+1) multiplied by itself five times. Expanding this expression means writing it out as a sum of terms. We can use the Binomial Theorem to do this efficiently.
The Binomial Theorem
The Binomial Theorem states:
(x + y)^n = ∑ (n choose k) * x^(n-k) * y^k
Where:
- n is the power of the binomial
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Applying the Binomial Theorem to (x+1)^5
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Identify n and y: In our case, n = 5 and y = 1.
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Calculate the binomial coefficients: We need to calculate (5 choose k) for k = 0, 1, 2, 3, 4, 5:
- (5 choose 0) = 5! / (0! * 5!) = 1
- (5 choose 1) = 5! / (1! * 4!) = 5
- (5 choose 2) = 5! / (2! * 3!) = 10
- (5 choose 3) = 5! / (3! * 2!) = 10
- (5 choose 4) = 5! / (4! * 1!) = 5
- (5 choose 5) = 5! / (5! * 0!) = 1
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Substitute the values into the Binomial Theorem:
(x + 1)^5 = (5 choose 0) * x^(5-0) * 1^0 + (5 choose 1) * x^(5-1) * 1^1 + (5 choose 2) * x^(5-2) * 1^2 + (5 choose 3) * x^(5-3) * 1^3 + (5 choose 4) * x^(5-4) * 1^4 + (5 choose 5) * x^(5-5) * 1^5
- Simplify:
(x + 1)^5 = 1 * x^5 * 1 + 5 * x^4 * 1 + 10 * x^3 * 1 + 10 * x^2 * 1 + 5 * x^1 * 1 + 1 * x^0 * 1
- Final Expansion:
(x + 1)^5 = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1
Conclusion
Using the Binomial Theorem, we have successfully expanded (x + 1)^5 into a polynomial expression. This method can be applied to expand any binomial raised to a positive integer power.