Conner Olson
09/14/2024 · Primary School

Use the Binomial Theorem to find the fourth term in the expansion of \( (x-2)^{8} \)

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To find the fourth term in the expansion of \( (x - 2)^8 \) using the Binomial Theorem, we start by recalling the theorem itself. The Binomial Theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In our case, we can identify \( a = x \), \( b = -2 \), and \( n = 8 \). The general term in the expansion can be expressed as: \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \] For our specific case, the \( k \)-th term (where \( k \) starts from 0) is given by: \[ T_{k+1} = \binom{8}{k} x^{8-k} (-2)^k \] To find the fourth term, we need to set \( k = 3 \) (since \( k \) starts from 0, the fourth term corresponds to \( k = 3 \)): \[ T_{4} = \binom{8}{3} x^{8-3} (-2)^3 \] Now, we calculate each part: 1. Calculate \( \binom{8}{3} \): \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \] 2. Calculate \( x^{8-3} \): \[ x^{8-3} = x^5 \] 3. Calculate \( (-2)^3 \): \[ (-2)^3 = -8 \] Now, we can combine these results to find the fourth term: \[ T_{4} = 56 \cdot x^5 \cdot (-8) = 56 \cdot (-8) \cdot x^5 = -448 x^5 \] Thus, the fourth term in the expansion of \( (x - 2)^8 \) is: \[ \boxed{-448 x^5} \]

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The fourth term in the expansion of \( (x - 2)^8 \) is \(-448 x^5\).
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