Understanding the Coefficient of ( x^9 ) in the Expansion of ( x^3 cdot frac{1}{x^{11}} )

In this article, we will explore how to find the coefficient of the term ( x^9 ) in the expansion of ( x^3 cdot frac{1}{x^{11}} ) using the binomial theorem, step-by-step. The binomial theorem provides a straightforward method to expand such expressions, making it easier to locate the desired term's coefficient.

The Binomial Theorem Explained

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form ( (a b)^n ) into a series of terms. For our specific expression, ( x^3 cdot frac{1}{x^{11}} ), we can utilize this theorem to identify the term with ( x^9 ).

Applying the Binomial Theorem

We begin by rewriting the expression in a form that facilitates the application of the binomial theorem:

Let ( a x^3 ) and ( b frac{1}{x^{11}} ), then the expansion of ( (a b)^n ) follows:

[ a b ]^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k

In our case, ( n 11 ), so we write:

( x^3 cdot frac{1}{x^{11}} sum_{k0}^{11} binom{11}{k} x^{3(11-k)} cdot frac{1}{x^k} )

This simplifies to:

( x^{33 - 3k - k} sum_{k0}^{11} binom{11}{k} x^{33 - 4k} )

Identifying the Desired Term

To find the term with ( x^9 ), we set the exponent of ( x ) equal to 9:

( 33 - 4k 9 )

By solving for ( k ), we get:

( 33 - 9 4k 24 4k k 6 )

Substituting ( k 6 ) back into the binomial coefficient:

( binom{11}{6} frac{11!}{6! cdot 5!} )

Calculating the Binomial Coefficient

Using the factorials:

( 11! 11 times 10 times 9 times 8 times 7 times 6! ) ( 6! 720 ) ( 5! 120 )

The binomial coefficient simplifies to:

( binom{11}{6} frac{11 times 10 times 9 times 8 times 7}{5 times 4 times 3 times 2 times 1} frac{55440}{120} 462 )

Therefore, the coefficient of the term ( x^9 ) in the expansion is ( boxed{462} ).

Verifying the Calculation

To verify, we can use specific values of ( x ). For example, if ( x 2 ):

( 2^{15} div 2^6 512 2^9 )

If ( x 3 ):

( 3^{15} div 3^6 19683 3^9 )

These results affirm the accuracy of the coefficients calculated using the binomial theorem.

Conclusion

The process of finding the coefficient of ( x^9 ) in the expansion of ( x^3 cdot frac{1}{x^{11}} ) involves applying the binomial theorem, setting the exponent of ( x ) to 9, solving for ( k ), and then calculating the binomial coefficient. This method ensures a systematic and accurate approach to polynomial expansions and coefficient extraction.