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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 49
Capitolo 9, Problema 49

Use the Binomial Theorem to expand each expression and write the result in simplified form. (x3 +x-2)4

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Identify the binomial expression to be expanded: \(\left(x^{3} + x^{-2}\right)^4\).
Recall the Binomial Theorem formula: \(\left(a + b\right)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), where \(\binom{n}{k}\) is the binomial coefficient.
Apply the formula with \(a = x^{3}\), \(b = x^{-2}\), and \(n = 4\). Write the sum as \(\sum_{k=0}^{4} \binom{4}{k} (x^{3})^{4-k} (x^{-2})^{k}\).
Simplify each term inside the sum by applying the power of a power rule: \((x^{3})^{4-k} = x^{3(4-k)}\) and \((x^{-2})^{k} = x^{-2k}\). Then combine the powers of \(x\) by adding exponents: \(x^{3(4-k) + (-2k)}\).
Write out each term explicitly for \(k=0\) to \(k=4\) using the binomial coefficients \(\binom{4}{k}\), simplify the powers of \(x\), and then sum all terms to get the expanded expression.

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It states that (a + b)^n equals the sum of terms C(n, k) * a^(n-k) * b^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the expression repeatedly.
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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 found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!). These coefficients determine the weight of each term in the expansion.
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Special Products - Cube Formulas

Handling Negative and Fractional Exponents

When expanding expressions like (x^3 + x^-2)^4, it is important to correctly apply exponent rules. Negative exponents indicate reciprocals (x^-2 = 1/x^2), and when multiplying powers with the same base, exponents add. Simplifying each term after expansion requires careful management of these exponents to write the final expression in simplest form.
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Zero and Negative Rules
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