How to Find the Constant Term in the Expansion of (1 3x^5)(1 3/x - 1)^2

In mathematical analysis, it is often required to find the constant term in the expansion of polynomial expressions. This article delves into the process of finding the constant term in the expansion of (1 3x^5)(1 3/x - 1)^2 using the binomial theorem. The binomial theorem is a powerful tool in algebraic expansions, enabling us to break down complex expressions into simpler components. Let's explore this step-by-step.

Step 1: Expand Each Factor Using the Binomial Theorem

The first step is to expand each part of the expression separately. We start with the factor (1 3x^5).

For the expression (1 3x^5):

1 3x^5 sum_{k0}^{5} binom{5}{k} 3^k x^k

Expanding it, we get:

1 3x 9x^2 27x^3 81x^4 243x^5

Next, we expand the second factor (1 3/x - 1)^2 which simplifies to (1 3/x - 1)^2 (3/x)^2.

For the expression (3/x)^2:

(1 3/x - 1)^2 sum_{j0}^{2} binom{2}{j} (3/x)^j 1^{2-j}

Simplifying, we get:

binom{2}{0} binom{2}{1} (3/x) binom{2}{2} (3/x)^2

1 6(1/x) 9(1/x^2)

Step 2: Combine the Expansions

Now that we have the expanded forms of both factors, we combine them to get the full expression:

1 3x^5 (1 6(1/x) 9(1/x^2))

Step 3: Find the Constant Term

The constant term in the combined expression is a term where the powers of x add up to zero. We look for pairs (k, j) such that:

k - j 0, therefore k j

For the factor (1 3x^5), the relevant power is k 0, 1, 2.

For the factor (1 6(1/x) 9(1/x^2)), the relevant power is j 0, 1, 2.

The relevant pairs (k, j) are (0, 0), (1, 1), and (2, 2).

Calculate Each Constant Term

For k 0 and j 0:

Term binom{5}{0} 3^0 * binom{2}{0} 3^0 1 * 1 1

For k 1 and j 1:

Term binom{5}{1} 3^1 * binom{2}{1} 3^1 5 * 3 * 2 * 3 90

For k 2 and j 2:

Term binom{5}{2} 3^2 * binom{2}{2} 3^2 10 * 9 * 1 * 9 810

Step 4: Sum the Constant Terms

The constant term in the final expression is the sum of all the constant terms we calculated:

1 90 810 901

Conclusion

The constant term in the expansion of (1 3x^5)(1 3/x - 1)^2 is 901. This process demonstrates the utility of the binomial theorem in simplifying and finding specific terms in polynomial expansions.

Understanding this method is crucial for those working with algebraic expressions and polynomials. By breaking down the problem into manageable steps, we can systematically find the constant term in any given polynomial expansion.