Long Division of Polynomials: Dividing (x^3 - x) by (x - 1)

Polynomial long division is a crucial algebraic tool that allows us to simplify complex expressions by dividing them into simpler components. This article will walk you through the process of dividing (x^3 - x) by (x - 1), using the method of long division. By the end of this guide, you will be able to perform similar divisions with greater confidence.

Understanding the Problem

Let's start with the polynomial expression (x^3 - x). We need to divide it by (x - 1) to find the quotient and the remainder. This can be written as:

[ frac{x^3 - x}{x - 1} ]

To divide the polynomial (x^3 - x) by (x - 1), we set up the division in a format similar to long division in arithmetic. The steps involved in polynomial long division are analogous to the steps in arithmetic long division, which makes it easier to follow along.

Step-by-Step Guide: Performing the Division

Step 1: Write down the dividend (x^3 - x) and the divisor (x - 1) in the long division format.

Arrange them as:

(x^3 - x) 0

Step 2: Determine how many times the first term of the divisor ((x)) can go into the first term of the dividend ((x^3)).

(x^3) divided by (x) is (x^2).

Write (x^2) as the first term of the quotient.

Step 3: Multiply the divisor (x - 1) by (x^2) and subtract the result from the original polynomial.

Multiplying (x^2) by (x - 1) gives us (x^3 - x^2).

Subtract (x^3 - x^2) from (x^3 - x 0).

[ x^3 - x 0 - (x^3 - x^2) x^2 - x 0 ]

Step 4: Bring down the next term from the dividend, which is (0).

Step 5: Determine how many times the first term of the divisor ((x)) can go into the new polynomial (x^2 - x 0).

(x^2) divided by (x) is (x).

Write (x) as the second term of the quotient.

Step 6: Multiply the divisor (x - 1) by (x) and subtract the result from the polynomial.

Multiplying (x) by (x - 1) gives us (x^2 - x).

Subtract (x^2 - x) from (x^2 - x 0).

[ x^2 - x 0 - (x^2 - x) 0 0 0 ]

Step 7: Determine how many times the first term of the divisor ((x)) can go into the new polynomial (2x 0).

Since (2x) is not divisible by (x), the next term in the quotient is (2).

Step 8: Multiply the divisor (x - 1) by (2) and subtract the result from the polynomial.

Multiplying (2) by (x - 1) gives us (2x - 2).

Subtract (2x - 2) from (2x 0).

[ 2x 0 - (2x - 2) 2 ]

The remainder is (2).

The Final Result

The quotient is (x^2 x 2) and the remainder is (2). Therefore, the result of the division is:

[frac{x^3 - x}{x - 1} x^2 x 2 frac{2}{x - 1}]

This means that:

[x^3 - x (x - 1)(x^2 x 2) 2]

Practical Applications

Understanding polynomial division is essential in various fields such as engineering, physics, and computer science. It helps in simplifying complex equations, solving polynomial equations, and understanding the behavior of functions. For example, in Calculus, polynomial division can be used to find asymptotes of rational functions.

Conclusion

Polynomial long division is a fundamental skill in algebra that allows us to break down complex expressions into simpler parts. By following the steps outlined in this article, you can now confidently perform long division on polynomials like (x^3 - x) by (x - 1). Practice with similar examples to reinforce your understanding and improve your algebraic skills.

Additional Resources

If you need further assistance or more practice problems, you can explore the following resources:

MathIsFun: Polynomials Wikipedia: Polynomial Long Division Khan Academy: Polynomial Long Division