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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 41
Capitolo 9, Problema 41

Find the term indicated in each expansion. (x − 1)9; fifth term

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Identify the general term formula for the binomial expansion of \((x - 1)^9\). The \(k\)-th term is given by \(T_{k+1} = \binom{9}{k} x^{9-k} (-1)^k\) where \(k\) starts from 0.
Since we want the fifth term, set \(k = 4\) because the first term corresponds to \(k=0\).
Substitute \(k=4\) into the general term formula: \(T_5 = \binom{9}{4} x^{9-4} (-1)^4\).
Calculate the binomial coefficient \(\binom{9}{4}\) using the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), but do not compute the numerical value yet.
Write the fifth term as \(T_5 = \binom{9}{4} x^5 (1)\) since \((-1)^4 = 1\), and this expression represents the fifth term in the expansion.

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n. It states that the expansion is the sum of terms involving binomial coefficients, powers of a, and powers of b. This theorem is essential for finding specific terms in binomial expansions.
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Special Products - Cube Formulas

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term.
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Term Identification in Binomial Expansion

In the expansion of (x + y)^n, the (k+1)-th term is given by C(n, k) * x^(n-k) * y^k. To find the fifth term, substitute k = 4. Understanding this formula allows you to directly compute any specific term without expanding the entire expression.
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Introduction to Polynomials
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