In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)^10

Verified step by step guidance

1. <insert step 1: Identify the binomial expression and the power. Here, the binomial is \((x - 2y)\) and the power is 10.>
2. <insert step 2: Use the Binomial Theorem, which states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), to expand the expression.>
3. <insert step 3: For the first term, set \(k = 0\): \(\binom{10}{0} (x)^{10} (-2y)^0\). Simplify this term.>
4. <insert step 4: For the second term, set \(k = 1\): \(\binom{10}{1} (x)^{9} (-2y)^1\). Simplify this term.>
5. <insert step 5: For the third term, set \(k = 2\): \(\binom{10}{2} (x)^{8} (-2y)^2\). Simplify this term.>

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n can be expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient. This theorem is essential for determining the coefficients and terms in the expansion of binomials.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', represent the number of ways to choose k elements from a set of n elements without regard to the order of selection. They can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients are crucial in the expansion of binomials as they determine the weight of each term in the expansion.

Simplification of Expressions

Simplification involves reducing expressions to their simplest form by combining like terms and eliminating unnecessary components. In the context of binomial expansions, this means organizing the terms produced by the expansion and ensuring that they are expressed in a clear and concise manner. This step is vital for presenting the final result in a way that is easy to interpret and use.