Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)⁴

Verified step by step guidance

1. Identify the binomial expression to be expanded: \((x^{2} + 2y)^4\).
2. Recall the Binomial Theorem formula: \(\displaystyle (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Set \(a = x^{2}\), \(b = 2y\), and \(n = 4\). Write the expansion as \(\sum_{k=0}^{4} \binom{4}{k} (x^{2})^{4-k} (2y)^k\).
4. Calculate each term by evaluating the binomial coefficient \(\binom{4}{k}\), raising \(x^{2}\) to the power \((4-k)\), and raising \$2y\( to the power \)k\(. Remember to simplify powers: \)(x^{2})^{m} = x^{2m}\( and \)(2y)^k = 2^k y^k$.
5. Write out all five terms from \(k=0\) to \(k=4\), simplify each term by multiplying coefficients and combining like terms, and then sum them to express the full expanded form.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It uses binomial coefficients, often represented by combinations, to determine the coefficients of each term in the expansion. This theorem simplifies the process of expanding powers of binomials without direct multiplication.

Binomial Coefficients and Combinations

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n without regard to order. These coefficients appear as the multipliers in each term of the binomial expansion and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!).

Simplifying Algebraic Expressions

After applying the Binomial Theorem, each term in the expansion may involve powers of variables and constants. Simplifying involves combining like terms, applying exponent rules (such as (x^a)^b = x^(ab)), and reducing coefficients to express the final expanded form clearly and concisely.