(x+2)^4 Binomial Theorem

Jasonbradley

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Jun 16, 2024

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Expanding (x+2)^4 using the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. This theorem is a powerful tool in algebra, allowing us to expand complex expressions without having to multiply them out manually.

Understanding the Binomial Theorem

The Binomial Theorem states:

(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 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 k = n.

Applying the Binomial Theorem to (x+2)^4

Let's expand (x+2)^4 using the binomial theorem:

1. Identify n: In this case, n = 4.

2. Expand the summation: We need to calculate the terms for k = 0, 1, 2, 3, and 4.

3. Calculate the 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

4. Substitute and simplify:

(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

(x + 2)^4 = 1 * x^4 * 1 + 4 * x^3 * 2 + 6 * x^2 * 4 + 4 * x * 8 + 1 * 1 * 16

**(x + 2)^4 = ** x^4 + 8x^3 + 24x^2 + 32x + 16

Conclusion

Therefore, the expanded form of (x+2)^4 using the binomial theorem is x^4 + 8x^3 + 24x^2 + 32x + 16. This demonstrates the power of the binomial theorem in simplifying complex algebraic expressions.