Finding the Constant k When the Coefficient of x3 in a Binomial Expansion is Given

In this article, we will explore how to find the constant k in a binomial expansion when the coefficient of x3 is known. We will employ the binomial theorem and basic algebraic manipulation to solve for k.

Introduction

The binomial theorem is a powerful tool for expanding expressions of the form (a b)n. This theorem is especially useful in expanding complex expressions and finding specific coefficients in the expansion. The given problem requires us to find the constant k such that the coefficient of x3 in the expansion of (1 kx)(1 - 2x)5 is 20. We will break down the solution step by step to ensure clarity.

Step-by-Step Solution

Expansion of (1 - 2x)5

Let's start by expanding (1 - 2x)5 using the binomial theorem:

(1 - 2x)5 sum;n05 binom{5}{n} (-2x)n

Expanding this, we get:

(1 - 2x)5 1 - 1 42 - 83 84 - 32x5

Combined Expansion (1 kx)(1 - 2x)5

Now, we need to find the coefficient of x3 in the expansion of (1 kx)(1 - 2x)5. To do this, we will multiply (1 kx) by each term in the expansion of (1 - 2x)5 and look at the resulting terms that contribute to x3.

The expansion of (1 kx)(1 - 2x)5 is:

(1 kx)(1 - 1 42 - 83 84 - 32x5)

This can be further expanded by distributing (1 kx) over each term:

(1 - 1 42 - 83 84 - 32x5) (kx - 10kx2 40kx3 - 80kx4 80kx5)

Combining like terms, the term containing x3 will be:

-83 40kx3

So, the coefficient of x3 is -80 40k.

Solving for k

We are given that the coefficient of x3 is 20. Therefore, we set up the equation:

-80 40k 20

Solving for k:

40k 100

k 5/2

Conclusion

In conclusion, the value of the constant k is 2.5. This was determined by expanding (1 - 2x)5 and then combining the terms to find the coefficient of x3 in the expansion of (1 kx)(1 - 2x)5. By setting this coefficient equal to 20, we were able to solve for k.

Additional Resources

For further study on binomial expansions and polynomial coefficients, we recommend exploring the following resources:

A detailed explanation of the binomial theorem at Wikipedia. Practice problems and solutions on binomial expansions at Khan Academy's website.