Use the Binomial Theorem to find the coefficient of x' in the expansion of (x+3)" ...-.
Use the Binomial Theorem to find the coefficient of x' in the expansion of (x+3)" ...-.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
This has to be a whole number
![**Exercise: Binomial Theorem Application**
**Objective:** Use the Binomial Theorem to find the coefficient of \(x^7\) in the expansion of \((x + 3)^{11}\).
\[ \underline{\hspace{20pt}} \]
□ \(x^7\)
**Explanation:**
To solve this problem, apply the Binomial Theorem, which is expressed as:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
For this specific case:
- \(x\) is \(x\) and \(y\) is \(3\).
- The exponent \(n\) is \(11\).
To find the coefficient of \(x^7\):
1. Set \(n-k = 7\) because we're interested in \(x^7\).
2. Solve for \(k\): \(k = n - 7 = 11 - 7 = 4\).
So we're finding the term where \(k = 4\):
\[
\binom{11}{4} x^{7} \cdot 3^4
\]
\(\binom{11}{4}\) is calculated as:
\[
\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330
\]
Calculate \(3^4\):
\[
3^4 = 81
\]
Thus, the coefficient of \(x^7\) is:
\[
330 \times 81 = 26,730
\]
Therefore, the coefficient of \(x^7\) in the expansion of \((x + 3)^{11}\) is **26,730**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72c2cbed-dcf9-4d9b-9c7c-d7c84e72018f%2F89202e98-f2c2-4774-980e-2526ab7b6e15%2Fuz6s79r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise: Binomial Theorem Application**
**Objective:** Use the Binomial Theorem to find the coefficient of \(x^7\) in the expansion of \((x + 3)^{11}\).
\[ \underline{\hspace{20pt}} \]
□ \(x^7\)
**Explanation:**
To solve this problem, apply the Binomial Theorem, which is expressed as:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
For this specific case:
- \(x\) is \(x\) and \(y\) is \(3\).
- The exponent \(n\) is \(11\).
To find the coefficient of \(x^7\):
1. Set \(n-k = 7\) because we're interested in \(x^7\).
2. Solve for \(k\): \(k = n - 7 = 11 - 7 = 4\).
So we're finding the term where \(k = 4\):
\[
\binom{11}{4} x^{7} \cdot 3^4
\]
\(\binom{11}{4}\) is calculated as:
\[
\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330
\]
Calculate \(3^4\):
\[
3^4 = 81
\]
Thus, the coefficient of \(x^7\) is:
\[
330 \times 81 = 26,730
\]
Therefore, the coefficient of \(x^7\) in the expansion of \((x + 3)^{11}\) is **26,730**.
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