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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 49
Chapter 9, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
Textbook Question

Express each repeating decimal as a fraction in lowest terms. 0.47‾=47100+4710,000+471,000,000+⋯0.\(\overline{47}\)=\(\frac{47}{100}\)+\(\frac{47}{10,000}\)+\(\frac{47}{1,000,000}\)+\(\cdots\)

Textbook Question

Find the term indicated in each expansion. (x2 + y)22; the term containing y14

Textbook Question

Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

Textbook Question

Express each repeating decimal as a fraction in lowest terms. 0.257‾0.\(\overline{257}\)

Textbook Question

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

∑i=11004i\(\sum\)_{i=1}^{100} 4i

Textbook Question

In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

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