How to Find the First Three Terms in Descending Powers of X in the Expansion of ( (x - frac{2}{x})^5 )
When working with polynomial expansions, the binomial theorem plays a crucial role. This theorem helps expand expressions of the form ( (a b)^n ) into a sum of terms of the form ( binom{n}{k} a^{n-k} b^k ). In this article, we will demonstrate how to apply the binomial theorem to find the first three terms in descending powers of x in the expansion of ( (x - frac{2}{x})^5 ).
The Binomial Theorem: An Overview
The binomial theorem states that any power of a binomial can be expanded as a sum of terms. For the expression ( (x - frac{2}{x})^5 ), we can apply the binomial theorem as follows:
( (x - frac{2}{x})^5 sum_{k0}^{5} binom{5}{k} x^{5-k} left(-frac{2}{x}right)^k )
Expansion of ( (x - frac{2}{x})^5 )
Let's break down the expansion step-by-step using the binomial theorem. We start by identifying the components of the binomial expression:
a x b -frac{2}{x} n 5Using the formula, we can generate the expansion:
( (x - frac{2}{x})^5 binom{5}{0} x^{5-0} left(-frac{2}{x}right)^0 binom{5}{1} x^{5-1} left(-frac{2}{x}right)^1 binom{5}{2} x^{5-2} left(-frac{2}{x}right)^2 )
Calculating the First Three Terms
Let's calculate each of the first three terms:
For ( k 0 ) Term: ( binom{5}{0} x^{5-0} left(-frac{2}{x}right)^0 1 cdot x^5 cdot 1 x^5 ) For ( k 1 ) Term: ( binom{5}{1} x^{5-1} left(-frac{2}{x}right)^1 5 cdot x^4 cdot left(-frac{2}{x}right) -1^3 ) For ( k 2 ) Term: ( binom{5}{2} x^{5-2} left(-frac{2}{x}right)^2 10 cdot x^3 cdot frac{4}{x^2} 4 )Combining these terms, we get:
( x^5 - 1^3 4 )
Conclusion
Thus, the first three terms in descending powers of x in the expansion of ( (x - frac{2}{x})^5 ) are:
( x^5 - 1^3 4 )
Welcome to the Full Expansion
For completeness, here is the full expansion:
( (x - frac{2}{x})^5 x^5 - 1^3 4 - 80 frac{80}{x} - frac{32}{x^3} )
This expansion includes all the terms generated using the binomial theorem.
Additional Key Information
The binomial coefficients for ( n 5 ) are as follows:
n Binomial Coefficients 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1Using the binomial coefficients, you can easily expand any polynomial expression according to the binomial theorem.