Binomial Expansion Formula Example
The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. It states:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
Where (n choose k) represents the binomial coefficient, which is calculated as:
(n choose k) = n! / (k! * (n-k)!)
Let's illustrate this with an example:
Example: Expanding (x + 2)^4
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Identify n: In this case, n = 4.
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Apply the binomial theorem:
(x + 2)^4 = ∑_(k=0)^4 (4 choose k) * x^(4-k) * 2^k
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Expand the summation:
(x + 2)^4 = (4 choose 0) * x^4 * 2^0 + (4 choose 1) * x^3 * 2^1 + (4 choose 2) * x^2 * 2^2 + (4 choose 3) * x^1 * 2^3 + (4 choose 4) * x^0 * 2^4
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Calculate binomial coefficients:
- (4 choose 0) = 4! / (0! * 4!) = 1
- (4 choose 1) = 4! / (1! * 3!) = 4
- (4 choose 2) = 4! / (2! * 2!) = 6
- (4 choose 3) = 4! / (3! * 1!) = 4
- (4 choose 4) = 4! / (4! * 0!) = 1
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Substitute the coefficients and simplify:
(x + 2)^4 = 1 * x^4 * 1 + 4 * x^3 * 2 + 6 * x^2 * 4 + 4 * x * 8 + 1 * 1 * 16
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Final result:
(x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16
This demonstrates the use of the binomial theorem to expand a binomial expression raised to a power. The process involves applying the formula, calculating the binomial coefficients, and simplifying the resulting terms.
