Write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)¹⁰

Verified step by step guidance

1. Identify the binomial expression and the exponent: here, the expression is \((x - 2y)^{10}\), where \(a = x\), \(b = -2y\), and \(n = 10\).
2. Recall the Binomial Theorem formula for expansion: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Write the first three terms by substituting \(k = 0, 1, 2\) into the formula: - For \(k=0\): \(\binom{10}{0} x^{10} (-2y)^0\) - For \(k=1\): \(\binom{10}{1} x^9 (-2y)^1\) - For \(k=2\): \(\binom{10}{2} x^8 (-2y)^2\).
4. Calculate each binomial coefficient: \(\binom{10}{0} = 1\), \(\binom{10}{1} = 10\), and \(\binom{10}{2} = 45\).
5. Simplify each term by applying the powers and multiplying coefficients, remembering to handle the negative sign in \((-2y)^k\) and express each term in simplified form.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum involving terms with binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the combination of n items taken k at a time.

Binomial Coefficients and Combinations

Binomial coefficients, denoted as C(n, k), represent the number of ways to choose k elements from n without regard to order. They are calculated using factorials: C(n, k) = n! / (k! * (n-k)!). These coefficients determine the multipliers for each term in the expansion.

Simplifying Terms in the Expansion

After applying the Binomial Theorem, each term must be simplified by calculating powers and multiplying coefficients. For example, powers of variables and constants are computed, and like terms are combined to express the expansion in its simplest form.