Simplifying Algebraic Expressions A Step-by-Step Guide

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It's like decluttering a room – you organize and streamline to reveal the essential elements. This article delves into the process of simplifying a given algebraic expression, providing a step-by-step guide to arrive at the most concise form. We'll break down the expression, factorize, identify common factors, and perform the necessary operations to reach the simplest representation. This journey will not only enhance your understanding of algebraic manipulation but also equip you with the tools to tackle more complex mathematical problems.

The Art of Simplifying Algebraic Expressions

At its core, simplifying an algebraic expression means rewriting it in a more compact and manageable form without altering its value. This often involves combining like terms, factoring polynomials, and canceling common factors. The goal is to present the expression in its most elegant and understandable form. This process is crucial in various mathematical contexts, from solving equations to calculus and beyond. Mastering simplification techniques is essential for anyone seeking to excel in mathematics and related fields.

The Given Expression A Starting Point

Let's begin with the expression we aim to simplify:

15x−35x2−11x+2+x−52x2−11x+5\frac{15 x-3}{5 x^2-11 x+2}+\frac{x-5}{2 x^2-11 x+5}

This expression involves two rational functions, each with a polynomial in the numerator and denominator. Our task is to combine these fractions and simplify the resulting expression as much as possible. This requires a systematic approach, involving factorization, finding common denominators, and simplifying the resulting numerator.

Factorization The Key to Simplification

The first step in simplifying this expression is to factorize the polynomials in the numerators and denominators. Factorization is the process of breaking down a polynomial into a product of simpler expressions (factors). This is a crucial step because it allows us to identify common factors that can be canceled out, leading to simplification.

Factoring the Denominators

Let's start by factoring the denominators:

  • First denominator: 5x^2 - 11x + 2

    To factor this quadratic, we look for two numbers that multiply to (5)(2) = 10 and add up to -11. These numbers are -10 and -1. We can then rewrite the middle term and factor by grouping:

    5x^2 - 10x - x + 2

    5x(x - 2) - 1(x - 2)

    (5x - 1)(x - 2)

  • Second denominator: 2x^2 - 11x + 5

    Similarly, we look for two numbers that multiply to (2)(5) = 10 and add up to -11. These numbers are -10 and -1. Rewriting the middle term and factoring by grouping:

    2x^2 - 10x - x + 5

    2x(x - 5) - 1(x - 5)

(2x - 1)(x - 5)

Factoring the Numerators

Now, let's factor the numerators:

  • First numerator: 15x - 3

    This is a simple linear expression. We can factor out a common factor of 3:

    3(5x - 1)

  • Second numerator: x - 5

    This expression is already in its simplest form and cannot be factored further.

Rewriting the Expression with Factored Forms

Now that we've factored the numerators and denominators, we can rewrite the original expression:

3(5x−1)(5x−1)(x−2)+(x−5)(2x−1)(x−5)\frac{3(5 x-1)}{(5 x-1)(x-2)}+\frac{(x-5)}{(2 x-1)(x-5)}

Identifying and Canceling Common Factors

Next, we look for common factors in the numerators and denominators that can be canceled out. This is a crucial step in simplifying rational expressions.

  • In the first term, we have a common factor of (5x - 1) in the numerator and denominator. Canceling this factor, we get:

    3(x−2)\frac{3}{(x-2)}

  • In the second term, we have a common factor of (x - 5) in the numerator and denominator. Canceling this factor, we get:

    1(2x−1)\frac{1}{(2 x-1)}

Combining the Simplified Fractions

Now our expression looks much simpler:

3(x−2)+1(2x−1)\frac{3}{(x-2)}+\frac{1}{(2 x-1)}

To add these fractions, we need a common denominator. The common denominator is the product of the two denominators: (x - 2)(2x - 1).

We rewrite each fraction with the common denominator:

3(2x−1)(x−2)(2x−1)+1(x−2)(2x−1)(x−2)\frac{3(2 x-1)}{(x-2)(2 x-1)}+\frac{1(x-2)}{(2 x-1)(x-2)}

Now we can combine the numerators:

3(2x−1)+1(x−2)(x−2)(2x−1)\frac{3(2 x-1)+1(x-2)}{(x-2)(2 x-1)}

Simplifying the Numerator

Let's simplify the numerator by distributing and combining like terms:

6x−3+x−2(x−2)(2x−1)\frac{6x - 3 + x - 2}{(x-2)(2 x-1)}

7x−5(x−2)(2x−1)\frac{7x - 5}{(x-2)(2 x-1)}

The Simplified Expression

Now we have:

7x−5(x−2)(2x−1)\frac{7 x-5}{(x-2)(2 x-1)}

We can expand the denominator, if desired:

7x−52x2−5x+2\frac{7 x-5}{2 x^2-5 x+2}

This is the simplest form of the given expression.

Conclusion Mastering Simplification Techniques

In this article, we've walked through the process of simplifying a complex algebraic expression. We've seen how factorization, identifying common factors, and combining fractions with common denominators are essential tools in this process. By mastering these techniques, you can confidently tackle a wide range of algebraic simplification problems. Remember, the key is to break down the problem into smaller, manageable steps and apply the appropriate techniques systematically. The simplified form of the expression $\frac{15 x-3}{5 x^2-11 x+2}+\frac{x-5}{2 x^2-11 x+5}$ is $\frac{7 x-5}{2 x^2-5 x+2}$, which corresponds to option A.

This journey into simplification underscores the importance of algebraic manipulation skills. These skills are not just confined to the classroom; they are applicable in various fields, including engineering, computer science, and economics. As you continue your mathematical journey, remember that simplification is a powerful tool for making complex problems more tractable and understandable.

What is the simplest form of the expression? $\frac{15 x-3}{5 x^2-11 x+2}+\frac{x-5}{2 x^2-11 x+5}$

Simplifying Algebraic Expressions Step-by-Step Solution