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].
