Domanda del libro di testo
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n2
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 9, Problema 11
Use the Binomial Theorem to expand each binomial and express the result in simplified form.
Guida verificata passo dopo passo1
Identify the binomial expression to expand: \((3x + y)^3\).
Recall the Binomial Theorem formula: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
Set \(a = 3x\), \(b = y\), and \(n = 3\). Write out each term of the expansion using the formula: \(\binom{3}{k} (3x)^{3-k} y^k\) for \(k = 0, 1, 2, 3\).
Calculate each binomial coefficient: \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\), and simplify the powers of \$3x$ and $y$ accordingly.
Write the full expanded expression by summing all terms and simplify each term by multiplying coefficients and combining like factors.
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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 involving binomial coefficients, powers of a, and powers of b. This theorem simplifies the expansion process without multiplying the binomial repeatedly.
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03:41
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 expanded form of (a + b)^n and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!).
Video consigliato:
03:41Special Products - Cube Formulas
Exponent Rules and Simplification
When expanding binomials, powers of each term must be calculated using exponent rules, such as (x^m)(x^n) = x^(m+n). After expansion, like terms should be combined and simplified to express the result in its simplest form, ensuring clarity and correctness.
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7:39Introduction to Exponent Rules
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