Expanding (x+1)^8
Expanding a binomial raised to a power can be tedious, but there's a powerful tool to help us: the Binomial Theorem. This theorem provides a formula to directly calculate each term in the expansion.
The Binomial Theorem
The Binomial Theorem states:
(x + y)^n = ∑(n choose k) * x^(nk) * y^k,
where:
 n is the power to which the binomial is raised.
 k ranges from 0 to n.
 (n choose k) represents the binomial coefficient, calculated as n! / (k! * (nk)!). This value represents the number of ways to choose k objects from a set of n.
Applying the Theorem to (x+1)^8
Let's apply the Binomial Theorem to expand (x+1)^8:

Identify n and y: In this case, n = 8 and y = 1.

Expand the summation: We need to calculate the terms for k = 0, 1, 2,... 8.

Calculate the binomial coefficients:
 (8 choose 0) = 1
 (8 choose 1) = 8
 (8 choose 2) = 28
 (8 choose 3) = 56
 (8 choose 4) = 70
 (8 choose 5) = 56
 (8 choose 6) = 28
 (8 choose 7) = 8
 (8 choose 8) = 1

Substitute the values into the theorem:
(x+1)^8 = (8 choose 0) * x^8 * 1^0 + (8 choose 1) * x^7 * 1^1 + (8 choose 2) * x^6 * 1^2 + (8 choose 3) * x^5 * 1^3 + (8 choose 4) * x^4 * 1^4 + (8 choose 5) * x^3 * 1^5 + (8 choose 6) * x^2 * 1^6 + (8 choose 7) * x^1 * 1^7 + (8 choose 8) * x^0 * 1^8
 Simplify:
(x+1)^8 = x^8 + 8x^7 + 28x^6 + 56x^5 + 70x^4 + 56x^3 + 28x^2 + 8x + 1
Conclusion
By applying the Binomial Theorem, we have successfully expanded (x+1)^8. This theorem is a powerful tool for efficiently expanding binomials raised to any power, saving us from tedious manual multiplication. Remember, understanding the formula and applying it step by step will help you confidently expand any binomial expression.