%5E6-expand.jpeg "(x+2)^6 Expand")
Expanding (x + 2)^6: A Guide to the Binomial Theorem
Expanding expressions like (x + 2)^6 can be tedious if done by hand. However, the Binomial Theorem provides a systematic way to expand such expressions. Here's a breakdown of how it works:
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) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This is often denoted as "nCk".
- ∑_(k=0)^n signifies the sum of the terms for values of k from 0 to n.
Applying the Binomial Theorem to (x + 2)^6
Let's apply this to expand (x + 2)^6:
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Identify 'n' and 'k': In this case, n = 6 (the power of the binomial). 'k' will range from 0 to 6.
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Calculate binomial coefficients: We'll need to calculate (6 choose k) for each value of k:
- (6 choose 0) = 6! / (0! * 6!) = 1
- (6 choose 1) = 6! / (1! * 5!) = 6
- (6 choose 2) = 6! / (2! * 4!) = 15
- (6 choose 3) = 6! / (3! * 3!) = 20
- (6 choose 4) = 6! / (4! * 2!) = 15
- (6 choose 5) = 6! / (5! * 1!) = 6
- (6 choose 6) = 6! / (6! * 0!) = 1
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Apply the formula: Substitute the values into the Binomial Theorem: (x + 2)^6 = (6 choose 0) * x^6 * 2^0 + (6 choose 1) * x^5 * 2^1 + (6 choose 2) * x^4 * 2^2 + (6 choose 3) * x^3 * 2^3 + (6 choose 4) * x^2 * 2^4 + (6 choose 5) * x^1 * 2^5 + (6 choose 6) * x^0 * 2^6
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Simplify: (x + 2)^6 = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64
Therefore, the expanded form of (x + 2)^6 is x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64.
Key Points
- The Binomial Theorem provides a structured method for expanding binomials raised to any power.
- Understanding binomial coefficients is essential for using the theorem.
- The expanded form of (x + 2)^6 illustrates how the powers of 'x' decrease while the powers of '2' increase as you move through the terms.
With practice, you can master the Binomial Theorem and expand any binomial expression efficiently.