Is There an (x^3) Term in the Expansion of (x^2 - frac{1}{x^7})?

When dealing with algebraic expressions, understanding whether certain terms appear in the expansion of a given expression is a common question. In the case of the expression (x^2 - frac{1}{x^7}), we want to determine if the expansion contains an (x^3) term. This article will walk through how to approach this problem step-by-step.

Using the Binomial Theorem

One method to solve this involves using the binomial theorem. The binomial theorem states that for any real numbers (a) and (b), and any non-negative integer (n), the expansion of ((a b)^n) is given by:

[ (a b)^n sum_{m 0}^{n} binom{n}{m} a^{n - m} b^m ]

In our problem, we want to find an (x^3) term in the expansion of ( left( x^2 - frac{1}{x} right)^7 ).

Step-by-Step Expansion

First, let's rewrite the expression using the binomial theorem:

[ left( x^2 - frac{1}{x} right)^7 sum_{k 0}^{7} binom{7}{k} x^{2(7 - k)} left( -frac{1}{x} right)^k ]

This expands as:

[ left( x^2 - frac{1}{x} right)^7 binom{7}{0} x^{14} - binom{7}{1} x^{12} frac{1}{x} binom{7}{2} x^{10} frac{1}{x^2} - binom{7}{3} x^8 frac{1}{x^3} cdots - binom{7}{7} x^0 frac{1}{x^7} ]

Simplifying the exponents, we get:

[ x^{14} - 7x^{11} 21x^8 - 35x^5 35x^2 - 21x^{-1} - frac{1}{x^7} ]

From this expansion, we can see that the exponents in the polynomial terms are (14, 11, 8, 5, 2, -1, -7). None of these exponents is (3), which means there is no (x^3) term in the expansion.

Understanding the Pattern

Another way to understand why there is no (x^3) term is to examine the numerator of the expanded form. We can rewrite the expression as:

[ left( x^2 - frac{1}{x} right)^7 left( x^3 - 1 right)^7 x^{-7} ]

The term (left( x^3 - 1 right)^7) expands to terms where the exponents of (x) are multiples of 3 (since only every third power of (x) contributes). Therefore, the expansion includes terms like (x^{21}), (x^{18}), (x^{15}), (x^{12}), etc. However, (x^3) is not among these terms.

Final Conclusion

Based on the binomial expansion and the pattern analysis, we can conclude that the expansion of (x^2 - frac{1}{x^7}) does not contain an (x^3) term.

Related Concepts

Understanding the behavior of exponents and the application of the binomial theorem are key to solving problems like this. Related concepts include:

Binomial Theorem: A formula for expanding powers of binomials. Exponentiation: Rules and properties of exponents. Polynomial Expansions: How to expand polynomial expressions.

By mastering these concepts, you can tackle similar algebraic problems with confidence.