Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Sure! Let's break down the process for expanding the expression [tex]\(\left(3 x^2 - \frac{1}{3 x}\right)^9\)[/tex] and finding the coefficient of [tex]\(x^6\)[/tex] in this expansion step by step.
1. Understanding the Expression:
The expression given is [tex]\(\left(3 x^2 - \frac{1}{3 x}\right)^9\)[/tex].
2. Binomial Expansion:
We can use the binomial theorem to expand [tex]\(\left(a + b\right)^n\)[/tex], where [tex]\(a = 3x^2\)[/tex], [tex]\(b = -\frac{1}{3x}\)[/tex] and [tex]\(n = 9\)[/tex].
3. Form of Terms in Binomial Expansion:
Each term in the expansion of [tex]\((a + b)^n\)[/tex] takes the form:
[tex]\[ \binom{n}{k} a^{n-k} b^{k} \][/tex]
In our expression:
[tex]\[ \binom{9}{k} (3x^2)^{9-k} \left(-\frac{1}{3x}\right)^{k} \][/tex]
4. Simplifying Each Term:
Each term [tex]\(\binom{9}{k} (3x^2)^{9-k} \left(-\frac{1}{3x}\right)^{k}\)[/tex] simplifies as follows:
- [tex]\(\binom{9}{k}\)[/tex] is the binomial coefficient.
- [tex]\((3x^2)^{9-k} = 3^{9-k} x^{2(9-k)}\)[/tex].
- [tex]\(\left(-\frac{1}{3x}\right)^k = (-1)^k \left(\frac{1}{3}\right)^k x^{-k}\)[/tex].
Combining these:
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot x^{2(9-k)} \cdot (-1)^k \cdot \left(\frac{1}{3}\right)^k \cdot x^{-k} \][/tex]
Simplify further:
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot (-1)^k \cdot \left(\frac{1}{3}\right)^k \cdot x^{18 - 3k} \][/tex]
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot \left(\frac{1}{3}\right)^k \cdot (-1)^k \cdot x^{18 - 3k} \][/tex]
[tex]\[ \binom{9}{k} \cdot 3^{9-2k} \cdot (-1)^k \cdot x^{18 - 3k} \][/tex]
5. Finding Coefficient for [tex]\(x^6\)[/tex]:
To find the coefficient of [tex]\(x^6\)[/tex], set the exponent of [tex]\(x\)[/tex] to 6:
[tex]\[ 18 - 3k = 6 \][/tex]
Solving for [tex]\(k\)[/tex]:
[tex]\[ 18 - 3k = 6 \Rightarrow 3k = 12 \Rightarrow k = 4 \][/tex]
6. Calculating the Specific Term:
Substitute [tex]\(k = 4\)[/tex] into the simplified term:
[tex]\[ \binom{9}{4} \cdot 3^{9-2\cdot4} \cdot (-1)^4 \cdot x^6 \][/tex]
[tex]\[ \binom{9}{4} \cdot 3^{9-8} \cdot 1 \cdot x^6 \][/tex]
[tex]\[ \binom{9}{4} \cdot 3^1 \cdot x^6 \][/tex]
The binomial coefficient [tex]\(\binom{9}{4} = 126\)[/tex].
Therefore, the coefficient is:
[tex]\[ 126 \cdot 3 = 378 \][/tex]
7. Conclusion:
The coefficient of the term [tex]\(x^6\)[/tex] in the expanded form of [tex]\(\left(3 x^2 - \frac{1}{3 x}\right)^9\)[/tex] is [tex]\( \boxed{378} \)[/tex].
1. Understanding the Expression:
The expression given is [tex]\(\left(3 x^2 - \frac{1}{3 x}\right)^9\)[/tex].
2. Binomial Expansion:
We can use the binomial theorem to expand [tex]\(\left(a + b\right)^n\)[/tex], where [tex]\(a = 3x^2\)[/tex], [tex]\(b = -\frac{1}{3x}\)[/tex] and [tex]\(n = 9\)[/tex].
3. Form of Terms in Binomial Expansion:
Each term in the expansion of [tex]\((a + b)^n\)[/tex] takes the form:
[tex]\[ \binom{n}{k} a^{n-k} b^{k} \][/tex]
In our expression:
[tex]\[ \binom{9}{k} (3x^2)^{9-k} \left(-\frac{1}{3x}\right)^{k} \][/tex]
4. Simplifying Each Term:
Each term [tex]\(\binom{9}{k} (3x^2)^{9-k} \left(-\frac{1}{3x}\right)^{k}\)[/tex] simplifies as follows:
- [tex]\(\binom{9}{k}\)[/tex] is the binomial coefficient.
- [tex]\((3x^2)^{9-k} = 3^{9-k} x^{2(9-k)}\)[/tex].
- [tex]\(\left(-\frac{1}{3x}\right)^k = (-1)^k \left(\frac{1}{3}\right)^k x^{-k}\)[/tex].
Combining these:
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot x^{2(9-k)} \cdot (-1)^k \cdot \left(\frac{1}{3}\right)^k \cdot x^{-k} \][/tex]
Simplify further:
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot (-1)^k \cdot \left(\frac{1}{3}\right)^k \cdot x^{18 - 3k} \][/tex]
[tex]\[ \binom{9}{k} \cdot 3^{9-k} \cdot \left(\frac{1}{3}\right)^k \cdot (-1)^k \cdot x^{18 - 3k} \][/tex]
[tex]\[ \binom{9}{k} \cdot 3^{9-2k} \cdot (-1)^k \cdot x^{18 - 3k} \][/tex]
5. Finding Coefficient for [tex]\(x^6\)[/tex]:
To find the coefficient of [tex]\(x^6\)[/tex], set the exponent of [tex]\(x\)[/tex] to 6:
[tex]\[ 18 - 3k = 6 \][/tex]
Solving for [tex]\(k\)[/tex]:
[tex]\[ 18 - 3k = 6 \Rightarrow 3k = 12 \Rightarrow k = 4 \][/tex]
6. Calculating the Specific Term:
Substitute [tex]\(k = 4\)[/tex] into the simplified term:
[tex]\[ \binom{9}{4} \cdot 3^{9-2\cdot4} \cdot (-1)^4 \cdot x^6 \][/tex]
[tex]\[ \binom{9}{4} \cdot 3^{9-8} \cdot 1 \cdot x^6 \][/tex]
[tex]\[ \binom{9}{4} \cdot 3^1 \cdot x^6 \][/tex]
The binomial coefficient [tex]\(\binom{9}{4} = 126\)[/tex].
Therefore, the coefficient is:
[tex]\[ 126 \cdot 3 = 378 \][/tex]
7. Conclusion:
The coefficient of the term [tex]\(x^6\)[/tex] in the expanded form of [tex]\(\left(3 x^2 - \frac{1}{3 x}\right)^9\)[/tex] is [tex]\( \boxed{378} \)[/tex].
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.