We appreciate your visit to Consider the binomial expression tex x y 10 tex Use the binomial theorem to complete each statement 1 The 3rd term of the expanded form. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
Sure! We can solve this question by using the binomial theorem step-by-step. The binomial theorem states that for any positive integer [tex]\( n \)[/tex],
[tex]\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, which can be calculated as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Let's consider the general term in the expansion of [tex]\( (x + y)^n \)[/tex]:
[tex]\[
T_{k+1} = \binom{n}{k} x^{n-k} y^k
\][/tex]
Following the given statements using the binomial expansion of [tex]\( (x + y)^{10} \)[/tex]:
1. Statement: The term of the expanded form of the given binomial expression is [tex]\( 45 x^8 y^2 \)[/tex].
- Here, we identify that [tex]\( n = 10 \)[/tex], [tex]\( x^{8} \)[/tex], and [tex]\( y^{2} \)[/tex].
- Comparing [tex]\(T_{k+1} = \binom{10}{k} x^{10-k} y^k\)[/tex] with [tex]\( 45 x^8 y^2 \)[/tex], we get:
[tex]\[
10 - k = 8 \quad \text{and} \quad k = 2
\][/tex]
[tex]\[
T_{3} = \binom{10}{2} x^8 y^2
\][/tex]
[tex]\[
\binom{10}{2} = \frac{10!}{2!8!} = 45
\][/tex]
- Therefore, [tex]\( k = 2 \)[/tex]. The third term [tex]\( T_3 \)[/tex] in expanded form indeed matches [tex]\( 45 x^8 y^2 \)[/tex].
2. Statement: The term of the expanded form of the given binomial expression is [tex]\( 252 x^5 y^5 \)[/tex].
- Here, we identify that [tex]\( x^5 \)[/tex] and [tex]\( y^5 \)[/tex].
- Comparing [tex]\( T_{k+1} = \binom{10}{k} x^{10-k} y^k \)[/tex] with [tex]\( 252 x^5 y^5 \)[/tex], we get:
[tex]\[
10 - k = 5 \quad \text{and} \quad k = 5
\][/tex]
[tex]\[
T_{6} = \binom{10}{5} x^5 y^5
\][/tex]
[tex]\[
\binom{10}{5} = \frac{10!}{5!5!} = 252
\][/tex]
- Therefore, [tex]\( k = 5 \)[/tex]. The sixth term [tex]\( T_6 \)[/tex] in expanded form indeed matches [tex]\( 252 x^5 y^5 \)[/tex].
3. Statement: The term of the expanded form of the given binomial expression is [tex]\( 10 x y^{\circ} \)[/tex].
- Here, we identify that [tex]\( x \)[/tex] and [tex]\( y^{\circ} \)[/tex] (assume this means [tex]\( y \)[/tex] with [tex]\( y^1 \)[/tex]).
- Comparing [tex]\( T_{k+1} = \binom{10}{k} x^{10-k} y^k \)[/tex] with [tex]\( 10 x y^1\)[/tex], we get:
[tex]\[
10 - k = 1 \quad \text{and} \quad k = 1
\][/tex]
[tex]\[
T_{2} = \binom{10}{1} x^9 y
\][/tex]
[tex]\[
\binom{10}{1} = \frac{10!}{1!9!} = 10
\][/tex]
- Therefore, [tex]\( k = 1 \)[/tex]. The second term [tex]\( T_2 \)[/tex] in the expanded form indeed matches [tex]\( 10 x y^1 \)[/tex].
In conclusion:
- The third term [tex]\( \binom{10}{2} x^8 y^2 \)[/tex] is [tex]\( 45 x^8 y^2 \)[/tex].
- The sixth term [tex]\( \binom{10}{5} x^5 y^5 \)[/tex] is [tex]\( 252 x^5 y^5 \)[/tex].
- The second term [tex]\( \binom{10}{1} x^9 y \)[/tex] is [tex]\( 10 x y \)[/tex].
[tex]\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, which can be calculated as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Let's consider the general term in the expansion of [tex]\( (x + y)^n \)[/tex]:
[tex]\[
T_{k+1} = \binom{n}{k} x^{n-k} y^k
\][/tex]
Following the given statements using the binomial expansion of [tex]\( (x + y)^{10} \)[/tex]:
1. Statement: The term of the expanded form of the given binomial expression is [tex]\( 45 x^8 y^2 \)[/tex].
- Here, we identify that [tex]\( n = 10 \)[/tex], [tex]\( x^{8} \)[/tex], and [tex]\( y^{2} \)[/tex].
- Comparing [tex]\(T_{k+1} = \binom{10}{k} x^{10-k} y^k\)[/tex] with [tex]\( 45 x^8 y^2 \)[/tex], we get:
[tex]\[
10 - k = 8 \quad \text{and} \quad k = 2
\][/tex]
[tex]\[
T_{3} = \binom{10}{2} x^8 y^2
\][/tex]
[tex]\[
\binom{10}{2} = \frac{10!}{2!8!} = 45
\][/tex]
- Therefore, [tex]\( k = 2 \)[/tex]. The third term [tex]\( T_3 \)[/tex] in expanded form indeed matches [tex]\( 45 x^8 y^2 \)[/tex].
2. Statement: The term of the expanded form of the given binomial expression is [tex]\( 252 x^5 y^5 \)[/tex].
- Here, we identify that [tex]\( x^5 \)[/tex] and [tex]\( y^5 \)[/tex].
- Comparing [tex]\( T_{k+1} = \binom{10}{k} x^{10-k} y^k \)[/tex] with [tex]\( 252 x^5 y^5 \)[/tex], we get:
[tex]\[
10 - k = 5 \quad \text{and} \quad k = 5
\][/tex]
[tex]\[
T_{6} = \binom{10}{5} x^5 y^5
\][/tex]
[tex]\[
\binom{10}{5} = \frac{10!}{5!5!} = 252
\][/tex]
- Therefore, [tex]\( k = 5 \)[/tex]. The sixth term [tex]\( T_6 \)[/tex] in expanded form indeed matches [tex]\( 252 x^5 y^5 \)[/tex].
3. Statement: The term of the expanded form of the given binomial expression is [tex]\( 10 x y^{\circ} \)[/tex].
- Here, we identify that [tex]\( x \)[/tex] and [tex]\( y^{\circ} \)[/tex] (assume this means [tex]\( y \)[/tex] with [tex]\( y^1 \)[/tex]).
- Comparing [tex]\( T_{k+1} = \binom{10}{k} x^{10-k} y^k \)[/tex] with [tex]\( 10 x y^1\)[/tex], we get:
[tex]\[
10 - k = 1 \quad \text{and} \quad k = 1
\][/tex]
[tex]\[
T_{2} = \binom{10}{1} x^9 y
\][/tex]
[tex]\[
\binom{10}{1} = \frac{10!}{1!9!} = 10
\][/tex]
- Therefore, [tex]\( k = 1 \)[/tex]. The second term [tex]\( T_2 \)[/tex] in the expanded form indeed matches [tex]\( 10 x y^1 \)[/tex].
In conclusion:
- The third term [tex]\( \binom{10}{2} x^8 y^2 \)[/tex] is [tex]\( 45 x^8 y^2 \)[/tex].
- The sixth term [tex]\( \binom{10}{5} x^5 y^5 \)[/tex] is [tex]\( 252 x^5 y^5 \)[/tex].
- The second term [tex]\( \binom{10}{1} x^9 y \)[/tex] is [tex]\( 10 x y \)[/tex].
Thanks for taking the time to read Consider the binomial expression tex x y 10 tex Use the binomial theorem to complete each statement 1 The 3rd term of the expanded form. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
