Simplifying Algebraic Expressions: A Guide to Dividing Polynomials

Algebraic expressions are a fundamental part of mathematics, and understanding how to simplify them is essential for students and professionals alike. This guide will walk you through the process of simplifying expressions involving polynomial division, focusing particularly on expressions like (frac{3x^2 - 4x - 5}{x - 1}).

Step-by-Step Simplification of Expressions

Consider the expression (frac{3x^2 - 4x - 5}{x - 1}). The goal here is to simplify this expression to make it easier to understand and work with. Let's break down the steps involved in simplifying this expression.

1. Breaking Down the Numerator

First, let's look at the numerator (3x^2 - 4x - 5). Notice that it can be factored into a part related to the denominator and a remainder.

Let's rewrite the expression as follows:

(frac{3x^2 - 4x - 5}{x - 1} frac{3x^2 - 3x - x - 5}{x - 1})

This allows us to separate the terms:

(frac{3x^2 - 3x - x - 5}{x - 1} frac{3x(x - 1) - (x 5)}{x - 1})

Notice that we can separate the terms in the numerator:

(frac{3x(x - 1) - (x 5)}{x - 1} frac{3x(x - 1)}{x - 1} - frac{x 5}{x - 1})

The first term simplifies to (3x), and we are left with a remainder:

(frac{3x(x - 1)}{x - 1} - frac{x 5}{x - 1} 3x - frac{x 5}{x - 1})

So, we have:

(frac{3x^2 - 4x - 5}{x - 1} 3x - frac{x 5}{x - 1})

2. Simplifying Further with Polynomial Division

Another way to approach the original expression is through polynomial long division. This method is particularly useful when the division does not result in a simple factorization.

Let's perform the polynomial long division.

Divide the leading term of the numerator by the leading term of the denominator: (frac{3x^2}{x} 3x)

Multiply the entire divisor by this result and subtract from the original numerator:

(3x^2 - 4x - 5 - (3x^2 - 3x) -x - 5)

Repeat the process with the new numerator: (frac{-x}{x} -1)

Multiply the entire divisor by this result and subtract from the new numerator:

(-x - 5 - (-x - 1) -4)

The final expression is:

(frac{3x^2 - 4x - 5}{x - 1} 3x - 1 - frac{4}{x - 1})

This leads us to:

(frac{3x^2 - 4x - 5}{x - 1} 3x - 1 - frac{4}{x - 1})

3. Final Simplified Form

Given the two methods, the simplified form of the expression (frac{3x^2 - 4x - 5}{x - 1}) is:

(frac{3x^2 - 4x - 5}{x - 1} 3x - 1 - frac{4}{x - 1})

Alternatively, if we want to express it as a single polynomial plus a remainder, it can be written as:

(frac{3x^2 - 4x - 5}{x - 1} 3x - 1 frac{-4}{x - 1})

or

(frac{3x^2 - 4x - 5}{x - 1} 3x - 1 frac{-4}{x - 1})

Conclusion

In conclusion, simplifying algebraic expressions involves a combination of factoring, polynomial long division, and careful separation of terms. Understanding these techniques is crucial for handling more complex mathematical problems. By applying these methods, you can make any polynomial expression more manageable and easier to work with.