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In how many ways can five airplanes line up for departure on a runway?
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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 82
Find the term in the expansion of (x2 + y2)5 containing x4 as a factor.
Guida verificata passo dopo passo1
Identify the general term in the binomial expansion of \(\left(x^{2} + y^{2}\right)^5\). The general term is given by \(T_{k+1} = \binom{5}{k} (x^{2})^{5-k} (y^{2})^{k}\), where \(k\) ranges from 0 to 5.
Simplify the powers in the general term: \(T_{k+1} = \binom{5}{k} x^{2(5-k)} y^{2k} = \binom{5}{k} x^{10 - 2k} y^{2k}\).
Since we want the term containing \(x^{4}\), set the exponent of \(x\) equal to 4: \(10 - 2k = 4\).
Solve for \(k\): \(10 - 2k = 4 \implies 2k = 6 \implies k = 3\).
Substitute \(k = 3\) back into the general term to find the specific term: \(T_{4} = \binom{5}{3} x^{10 - 2(3)} y^{2(3)} = \binom{5}{3} x^{4} y^{6}\).
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Binomial Theorem
The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum of terms involving binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the combination of n items taken k at a time. This theorem is essential for expanding (x² + y²)^5.
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Special Products - Cube Formulas
Binomial Coefficients
Binomial coefficients, denoted as C(n, k), represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal’s triangle. Understanding these coefficients helps identify the correct term in the expansion.
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03:41Special Products - Cube Formulas
Exponent Matching in Terms
To find a term containing a specific factor like x^4, you must match the exponents of variables in each term of the expansion. Since the expansion involves powers of x² and y², the exponents are multiples of 2. Identifying the term where the power of x² equals 4 (i.e., (x²)^2 = x^4) is key to solving the problem.
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7:39Introduction to Exponent Rules
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