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Magazine Chapter 1: Problem 8
What is the coefficient of \(x^{12}\) in \((x+2)^{15} ?\)
Short Answer
Expert verified
The coefficient of \(x^{12}\) in \( (x+2)^{15} \) is 3640.
Step by step solution
01
Identify the Expansion Formula
To find the coefficient of a specific term in a binomial expansion, we should use the Binomial Theorem, which states that \( (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \), where \({n \choose k}\) is the binomial coefficient.
02
Determine the Specific Term
The term that contains \(x^{12}\) will have the form \( {15 \choose k}x^{15-k}2^k \) for some integer \(k\). To find the coefficient, we need to solve for \(k\) when \(15-k=12\) which gives us \(k=3\).
03
Calculate the Binomial Coefficient
The binomial coefficient for \(k=3\) is \( {15 \choose 3} = \frac{15!}{3!(15-3)!} \).
04
Compute the Factor for \(2^k\)
The value of \(2^k\) when \(k=3\) is \(2^3 = 8\).
05
Find the Coefficient of \(x^{12}\)
The coefficient of \(x^{12}\) is the product of the binomial coefficient and the corresponding power of 2, which is \({15 \choose 3} \times 2^3\). Simplifying this product gives the coefficient.
06
Simplify the Calculation
Substitute the values to get \( {15 \choose 3} \times 2^3 = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} \times 8 = 5 \times 7 \times 13 \times 8 = 455 \times 8 = 3640\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
Understanding the binomial expansion is crucial for solving algebraic problems involving powers of binomials. A binomial is a mathematical expression with two terms, such as \(a + b\). The binomial theorem provides a formula for expanding expressions that are raised to a power, such as \(a + b)^n\).
The binomial theorem states that \(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\), where \(n\) is a non-negative integer, and \(a^{n-k} b^k\) denotes the terms in the expansion. Another crucial aspect to note is that the sum of the exponents in each term \(a^{n-k} b^k\) equals \(n\), the original power.
In our exercise, to find the specific term with \(x^{12}\), we used this theorem to expand \(x+2)^{15}\), and then located the term of interest.
The binomial theorem states that \(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\), where \(n\) is a non-negative integer, and \(a^{n-k} b^k\) denotes the terms in the expansion. Another crucial aspect to note is that the sum of the exponents in each term \(a^{n-k} b^k\) equals \(n\), the original power.
In our exercise, to find the specific term with \(x^{12}\), we used this theorem to expand \(x+2)^{15}\), and then located the term of interest.
Binomial Coefficient
The binomial coefficient, denoted as \( {n \choose k} \), is a central element in the binomial theorem and represents the number of ways to choose \(k\) elements from \(n\) different elements without considering the order. It is also referred to as \(\text{鈥渘 choose k鈥潁\) and can be calculated using the factorial function: \( {n \choose k} = \frac{n!}{k!(n-k)!} \).
In the context of the binomial expansion, it determines the multiplier for each term in the expansion. During the computation in our exercise, we calculated the binomial coefficient for \(k=3\) as \( {15 \choose 3} \) to find the coefficient of \(x^{12}\) in the expanded form of \(x+2)^{15}\). This step is pivotal for solving the problem as it directly affects the numerical value of the desired coefficient.
In the context of the binomial expansion, it determines the multiplier for each term in the expansion. During the computation in our exercise, we calculated the binomial coefficient for \(k=3\) as \( {15 \choose 3} \) to find the coefficient of \(x^{12}\) in the expanded form of \(x+2)^{15}\). This step is pivotal for solving the problem as it directly affects the numerical value of the desired coefficient.
Polynomial Coefficients
Polynomial coefficients are the numerical factors that multiply the variable terms in a polynomial expression. When expanding binomials, each term of the resulting polynomial has a coefficient determined by the binomial coefficients. These polynomial coefficients follow a pattern known as Pascal's triangle when looking at binomial expansions. In each term of the expansion \(a+b)^n\), the coefficient is the product of the binomial coefficient and the corresponding powers of \(a\) and \(b\).
For our exercise, after finding the binomial coefficient with \( {15 \choose 3} = 455 \), we then calculated \(2^3 = 8\) to determine the factor multiplying \(x^{12}\). The final coefficient of \(x^{12}\) was the product of these values, yielding 3640. This emphasizes how polynomial coefficients are integral to finding individual terms within a binomial expansion.
For our exercise, after finding the binomial coefficient with \( {15 \choose 3} = 455 \), we then calculated \(2^3 = 8\) to determine the factor multiplying \(x^{12}\). The final coefficient of \(x^{12}\) was the product of these values, yielding 3640. This emphasizes how polynomial coefficients are integral to finding individual terms within a binomial expansion.
