Answer:
[tex]336798\:x^9[/tex]
Step-by-step explanation:
Binomial Theorem
[tex]\displaystyle (a+b)^n=\binom{n}{0}a^{n-0}b^0+\binom{n}{1}a^{n-1}b^1+...+\binom{n}{r}a^{n-r}b^r+...+\binom{n}{n}a^{n-n}b^n[/tex]
[tex]\textsf{Where }\displaystyle \rm \binom{n}{r} \: = \:^{n}C_{r} = \frac{n!}{r!(n-r)!}[/tex]
Factorial is denoted by an exclamation mark "!" placed after the number. It means to multiply all whole numbers from the given number down to 1.
Example: 4! = 4 × 3 × 2 × 1
Given expression:
[tex]\left(x^3-\dfrac{3}{x}\right)^{11}[/tex]
Therefore:
[tex]\implies a=x^3[/tex]
[tex]\implies b=-\dfrac{3}{x}[/tex]
[tex]\implies n=11[/tex]
To find the value of r for the term in x⁹:
[tex]\implies a^{11-r} \cdot b^r=x^9[/tex]
[tex]\implies \dfrac{(x^3)^{11-r}}{x^r}=x^9[/tex]
[tex]\implies 3(11-r)-r=9[/tex]
[tex]\implies 33-4r=9[/tex]
[tex]\implies 4r=24[/tex]
[tex]\implies r=6[/tex]
Therefore, the term in x⁹ will be the 7th term, where r = 6:
[tex]\implies \displaystyle \binom{11}{6}a^{11-6}b^6[/tex]
[tex]\implies \displaystyle \dfrac{11!}{6!(11-6)!}a^{5}b^6[/tex]
[tex]\implies \displaystyle \dfrac{11!}{6!\:5!}a^{5}b^6[/tex]
[tex]\implies \dfrac{11 \times 10\times 9\times 8\times 7\times \diagup\!\!\!\!\!6\times \diagup\!\!\!\!\!5\times \diagup\!\!\!\!\!4\times \diagup\!\!\!\!\!3\times \diagup\!\!\!\!\!2\times \diagup\!\!\!\!\!1}{\diagup\!\!\!\!\!6 \times\diagup\!\!\!\!\!5 \times\diagup\!\!\!\!\!4 \times\diagup\!\!\!\!\!3 \times\diagup\!\!\!\!\!2 \times\diagup\!\!\!\!\!1 \times5 \times4 \times3 \times2 \times1}a^5b^6[/tex]
[tex]\implies \dfrac{55440}{120}a^5b^6[/tex]
[tex]\implies 462a^5b^6[/tex]
Substitute the values of a and b:
[tex]\implies 462\left(x^3\right)^5\left(-\dfrac{3}{x}\right)^6[/tex]
[tex]\implies 462\:x^{15}\left(-\dfrac{3^6}{x^6}\right)[/tex]
[tex]\implies 462\:x^{15}\left(\dfrac{729}{x^6}\right)[/tex]
[tex]\implies 462 \cdot 729\left(\dfrac{x^{15}}{x^6}\right)[/tex]
[tex]\implies 336798\:x^9[/tex]
Learn more about binomial expansion here:
https://brainly.com/question/27957648
