EX1 FIND THE 4th TERM BINOMIAL EXPANSION (3x+2y) ³ IN THE OF 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°.
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## Binomial Expansion: Finding a Specific Term
### Example 1: Find the 4th Term in the Binomial Expansion of (3x + 2y)⁵
1. **Problem Statement:**
\[
\text{Find the 4th term in the binomial expansion of } (3x + 2y)^5.
\]
2. **Solution:**
- The expansion follows the binomial theorem formula:
\[
\binom{n}{r} \cdot (a)^{n-r} \cdot (b)^r
\]
- For the given problem, \(n = 5\), \(a = 3x\), and \(b = 2y\).
- The 4th term corresponds to \(k = 3\) (since we start counting from 0).
\[
T_{k+1} = \binom{5}{3} \cdot (3x)^{5-3} \cdot (2y)^3
\]
- Calculating each part separately:
\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1)(2 \cdot 1)} = 10
\]
\[
(3x)^2 = 9x^2
\]
\[
(2y)^3 = 8y^3
\]
- Combining these results:
\[
10 \cdot 9x^2 \cdot 8y^3 = 720x^2y^3
\]
- Therefore, the 4th term is:
\[
\boxed{720x^2y^3}
\]
### Example 2: Find the 7th Term in the Binomial Expansion of (4x - y)⁷
1. **Problem Statement:**
\[
\text{Find the 7th term in the binomial expansion of } (4x - y)^7.
\]
### Additional Information:
- The binomial coefficient is computed using:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52259713-9524-4403-8816-50aefee31531%2F056f560b-6a4b-4110-b64d-6b5dbab25023%2Ftkztefl_processed.png&w=3840&q=75)
Transcribed Image Text:---
## Binomial Expansion: Finding a Specific Term
### Example 1: Find the 4th Term in the Binomial Expansion of (3x + 2y)⁵
1. **Problem Statement:**
\[
\text{Find the 4th term in the binomial expansion of } (3x + 2y)^5.
\]
2. **Solution:**
- The expansion follows the binomial theorem formula:
\[
\binom{n}{r} \cdot (a)^{n-r} \cdot (b)^r
\]
- For the given problem, \(n = 5\), \(a = 3x\), and \(b = 2y\).
- The 4th term corresponds to \(k = 3\) (since we start counting from 0).
\[
T_{k+1} = \binom{5}{3} \cdot (3x)^{5-3} \cdot (2y)^3
\]
- Calculating each part separately:
\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1)(2 \cdot 1)} = 10
\]
\[
(3x)^2 = 9x^2
\]
\[
(2y)^3 = 8y^3
\]
- Combining these results:
\[
10 \cdot 9x^2 \cdot 8y^3 = 720x^2y^3
\]
- Therefore, the 4th term is:
\[
\boxed{720x^2y^3}
\]
### Example 2: Find the 7th Term in the Binomial Expansion of (4x - y)⁷
1. **Problem Statement:**
\[
\text{Find the 7th term in the binomial expansion of } (4x - y)^7.
\]
### Additional Information:
- The binomial coefficient is computed using:
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