Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 19E
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![### Algebra 2B - GENET 21-22 (MI)
#### Module: Binomial Expansion
**Question 4:** What is the value of the 7th term in \((2x + y)^9\)?
**Options:**
a) \( \binom{9}{6} (2x)^{9-6} (y)^6 \)
b) \( \binom{9}{6} (a)^{9-6} (b)^6 \)
c) \( \binom{9}{6} (2x)^{9-6} (y)^6 \)
d) \( \binom{9}{6} (ax)^{9-6} (b)^{6-6} \)
**Explanation:**
To find the 7th term in the binomial expansion of \((2x + y)^9\), we can use the general term formula in the binomial theorem:
\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]
Given that \( n = 9 \), \( a = 2x \), and \( b = y \):
For the 7th term (\( k = 6 \)), we calculate as follows:
\[ T_{6+1} = \binom{9}{6} (2x)^{9-6} (y)^6 \]
which simplifies to:
\[ \binom{9}{6} (2x)^3 y^6 \]
By comparing this with the given options, we find that the correct option is:
\[ \boxed{ \binom{9}{6} (2x)^{9-6} (y)^6 } \]
(\(Note: This is both option (a) and option (c).\))
For deeper understanding, students can refer to the binomial theorem and practice expanding binomials for a variety of expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe20fc6e6-2ea7-4c1e-8366-712724b6961a%2F31dc0ed9-e260-4d1f-b45e-ba38c3c16220%2Ffjvqodh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebra 2B - GENET 21-22 (MI)
#### Module: Binomial Expansion
**Question 4:** What is the value of the 7th term in \((2x + y)^9\)?
**Options:**
a) \( \binom{9}{6} (2x)^{9-6} (y)^6 \)
b) \( \binom{9}{6} (a)^{9-6} (b)^6 \)
c) \( \binom{9}{6} (2x)^{9-6} (y)^6 \)
d) \( \binom{9}{6} (ax)^{9-6} (b)^{6-6} \)
**Explanation:**
To find the 7th term in the binomial expansion of \((2x + y)^9\), we can use the general term formula in the binomial theorem:
\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]
Given that \( n = 9 \), \( a = 2x \), and \( b = y \):
For the 7th term (\( k = 6 \)), we calculate as follows:
\[ T_{6+1} = \binom{9}{6} (2x)^{9-6} (y)^6 \]
which simplifies to:
\[ \binom{9}{6} (2x)^3 y^6 \]
By comparing this with the given options, we find that the correct option is:
\[ \boxed{ \binom{9}{6} (2x)^{9-6} (y)^6 } \]
(\(Note: This is both option (a) and option (c).\))
For deeper understanding, students can refer to the binomial theorem and practice expanding binomials for a variety of expressions.
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