Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 27
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)5
Verified step by step guidance1
Identify the binomial expression to expand: \((3x - y)^5\).
Recall the Binomial Theorem formula: \(\displaystyle (a - b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} (-b)^k\).
Set \(a = 3x\), \(b = y\), and \(n = 5\). Write the expansion as \(\displaystyle \sum_{k=0}^5 \binom{5}{k} (3x)^{5-k} (-y)^k\).
Calculate each term by finding the binomial coefficient \(\binom{5}{k}\), raising \$3x\( to the power \(5-k\), and \)-y\( to the power \)k$, then multiply all parts together.
Simplify each term by applying the powers and combining like terms, paying attention to the signs from \((-y)^k\), and write the full expanded polynomial.
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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 the expansion is the sum of terms involving binomial coefficients, powers of a, and powers of b. This theorem simplifies the process of expanding binomials without multiplying 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 Theorem expansion and can be found using Pascal's Triangle or the formula n! / (k!(n-k)!). These coefficients determine the weight of each term in the expansion.
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Exponent Rules in Expansion
When expanding a binomial like (3x − y)^5, each term involves powers of both 3x and −y. The exponents of these terms add up to the total power (5 in this case). Understanding how to apply exponent rules and distribute powers correctly is essential to simplify each term in the expansion.
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Related Practice
Textbook Question
Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. a1 = 9, d=2
Textbook Question
In Exercises 23–28, evaluate each factorial expression. 20!/2!18!
Textbook Question
Evaluate each factorial expression. (n+2)!/n!
Textbook Question
Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...
Textbook Question
Evaluate each expression.
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