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

검증된 단계별 안내

Identify the binomial expression and the exponent: here, the expression is \((x - 2y)^{10}\), where \(a = x\), \(b = -2y\), and \(n = 10\).

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.

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 \(k=2\): \(\binom{10}{2} x^8 (-2y)^2\).

Calculate each binomial coefficient: \(\binom{10}{0} = 1\), \(\binom{10}{1} = 10\), and \(\binom{10}{2} = 45\).

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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