Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of algebraic expressions and tackling a common challenge: simplifying them! Specifically, we're going to break down how to simplify the following expression:

15xโˆ’35x2โˆ’11x+2+xโˆ’52x2โˆ’11x+5\frac{15 x-3}{5 x^2-11 x+2}+\frac{x-5}{2 x^2-11 x+5}

Simplifying expressions like this might seem daunting at first, but don't worry! We'll take it one step at a time, making sure you understand the underlying concepts and can confidently simplify similar problems in the future. So, grab your pencils, and let's get started!

Understanding the Basics of Algebraic Expressions

Before we jump into the problem, let's quickly review what algebraic expressions are and why we simplify them. Algebraic expressions are combinations of variables (like x) and constants (numbers) connected by mathematical operations (+, -, ร—, รท). Simplifying an expression means rewriting it in a simpler, more compact form without changing its value. This makes it easier to work with and understand.

Think of it like this: Imagine you have a tangled-up ball of yarn. Simplifying the expression is like untangling the yarn, making it easier to see and use. In mathematics, simpler expressions are easier to analyze, solve equations, and graph functions. They help us to see the underlying relationships more clearly. So, simplification is not just about making things look prettier; it's about gaining a deeper understanding and making further calculations easier. Understanding these basics sets the stage for tackling more complex simplifications. We aim to make the expression as clean and manageable as possible. By mastering these techniques, you'll be well-equipped to handle more advanced algebraic manipulations.

Step 1: Factoring the Denominators

The first key step in simplifying this expression is to factor the denominators of both fractions. Factoring is like breaking down a number or expression into its multiplicative building blocks. It's crucial because it allows us to identify common factors, which we can then use to simplify the expression.

Let's start with the first denominator: 5x2โˆ’11x+25x^2 - 11x + 2. We need to find two binomials that multiply to give us this quadratic expression. This often involves a bit of trial and error, but there are some strategies we can use. We look for two numbers that multiply to (5)(2) = 10 and add up to -11. These numbers are -10 and -1. So, we can rewrite the middle term as -10x - x:

5x2โˆ’11x+2=5x2โˆ’10xโˆ’x+25x^2 - 11x + 2 = 5x^2 - 10x - x + 2

Now, we can factor by grouping:

5x2โˆ’10xโˆ’x+2=5x(xโˆ’2)โˆ’1(xโˆ’2)=(5xโˆ’1)(xโˆ’2)5x^2 - 10x - x + 2 = 5x(x - 2) - 1(x - 2) = (5x - 1)(x - 2)

Next, let's factor the second denominator: 2x2โˆ’11x+52x^2 - 11x + 5. We need two numbers that multiply to (2)(5) = 10 and add up to -11. Again, these numbers are -10 and -1. Rewrite the middle term:

2x2โˆ’11x+5=2x2โˆ’10xโˆ’x+52x^2 - 11x + 5 = 2x^2 - 10x - x + 5

And factor by grouping:

2x2โˆ’10xโˆ’x+5=2x(xโˆ’5)โˆ’1(xโˆ’5)=(2xโˆ’1)(xโˆ’5)2x^2 - 10x - x + 5 = 2x(x - 5) - 1(x - 5) = (2x - 1)(x - 5)

So, we've successfully factored both denominators. Remember, the goal here is to break down the expressions into simpler components, making it easier to find common factors and simplify the overall expression. Factoring correctly is essential for simplifying rational expressions. It's like laying the foundation for the rest of the simplification process. Without accurate factoring, the subsequent steps will likely lead to incorrect results. Make sure you practice factoring different types of quadratic expressions to become proficient in this skill. This step paves the way for combining the fractions by finding a common denominator.

Step 2: Rewrite the Expression with Factored Denominators

Now that we've factored the denominators, we can rewrite the original expression. This will give us a clearer picture of what we're working with and make it easier to identify common factors.

Our original expression was:

15xโˆ’35x2โˆ’11x+2+xโˆ’52x2โˆ’11x+5\frac{15 x-3}{5 x^2-11 x+2}+\frac{x-5}{2 x^2-11 x+5}

Using the factored forms of the denominators, we can rewrite this as:

15xโˆ’3(5xโˆ’1)(xโˆ’2)+xโˆ’5(2xโˆ’1)(xโˆ’5)\frac{15x - 3}{(5x - 1)(x - 2)} + \frac{x - 5}{(2x - 1)(x - 5)}

Notice how the factored form immediately highlights potential cancellations and simplifications. Before we move on, let's take a quick look at the numerators. In the first fraction, we have 15x - 3. We can factor out a 3 from this expression:

15xโˆ’3=3(5xโˆ’1)15x - 3 = 3(5x - 1)

Now, let's substitute this back into our expression:

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

Rewriting the expression in this form is crucial because it sets the stage for simplifying by canceling common factors. This is where we start to see the expression become less complex. The key here is to ensure that all parts of the expression are in their simplest factored form before proceeding. This step is like organizing your tools before starting a project; it makes the work much smoother and more efficient. By factoring and rewriting, we're preparing the expression for the next crucial step: finding a common denominator.

Step 3: Finding a Common Denominator

To add fractions, they need to have a common denominator. This is a fundamental principle of fraction arithmetic, and it applies to algebraic fractions as well. Looking at our rewritten expression:

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

We can see that the denominators are (5xโˆ’1)(xโˆ’2)(5x - 1)(x - 2) and (2xโˆ’1)(xโˆ’5)(2x - 1)(x - 5). The least common denominator (LCD) is the smallest expression that both denominators can divide into. In this case, the LCD is the product of all unique factors from both denominators:

LCD = (5xโˆ’1)(xโˆ’2)(2xโˆ’1)(xโˆ’5)(5x - 1)(x - 2)(2x - 1)(x - 5)

However, before we jump to this conclusion, let's take a closer look. We can actually simplify the expression before finding the common denominator by canceling common factors within each fraction. In the first fraction, we have (5xโˆ’1)(5x - 1) in both the numerator and denominator, and in the second fraction, we have (xโˆ’5)(x - 5) in both the numerator and denominator. So, we can cancel these out:

3(5xโˆ’1)(5xโˆ’1)(xโˆ’2)+(xโˆ’5)(2xโˆ’1)(xโˆ’5)=3xโˆ’2+12xโˆ’1\frac{3\cancel{(5x - 1)}}{\cancel{(5x - 1)}(x - 2)} + \frac{\cancel{(x - 5)}}{(2x - 1)\cancel{(x - 5)}} = \frac{3}{x - 2} + \frac{1}{2x - 1}

Now, our denominators are simply (xโˆ’2)(x - 2) and (2xโˆ’1)(2x - 1). This makes finding the common denominator much easier. The LCD is now just the product of these two factors:

LCD = (xโˆ’2)(2xโˆ’1)(x - 2)(2x - 1)

Finding the common denominator is a critical step in adding fractions. However, simplifying before finding the LCD, as we did here, can save a lot of work. It's like choosing the right tool for the job; simplifying first makes the entire process more efficient. Now that we have the LCD, we can rewrite each fraction with this denominator.

Step 4: Rewrite Fractions with the Common Denominator

Now that we have our common denominator, (xโˆ’2)(2xโˆ’1)(x - 2)(2x - 1), we need to rewrite each fraction with this denominator. This involves multiplying the numerator and denominator of each fraction by the factors needed to obtain the LCD.

Our simplified expression is:

3xโˆ’2+12xโˆ’1\frac{3}{x - 2} + \frac{1}{2x - 1}

For the first fraction, we need to multiply the numerator and denominator by (2xโˆ’1)(2x - 1):

3xโˆ’2โ‹…2xโˆ’12xโˆ’1=3(2xโˆ’1)(xโˆ’2)(2xโˆ’1)\frac{3}{x - 2} \cdot \frac{2x - 1}{2x - 1} = \frac{3(2x - 1)}{(x - 2)(2x - 1)}

For the second fraction, we need to multiply the numerator and denominator by (xโˆ’2)(x - 2):

12xโˆ’1โ‹…xโˆ’2xโˆ’2=1(xโˆ’2)(xโˆ’2)(2xโˆ’1)\frac{1}{2x - 1} \cdot \frac{x - 2}{x - 2} = \frac{1(x - 2)}{(x - 2)(2x - 1)}

Now, both fractions have the common denominator (xโˆ’2)(2xโˆ’1)(x - 2)(2x - 1). This allows us to add the fractions together. Rewriting fractions with a common denominator is a fundamental technique in fraction arithmetic. It's like speaking the same language; once both fractions are expressed in terms of the common denominator, we can combine them easily. This step sets the stage for the final simplification, where we combine the numerators and simplify the resulting expression.

Step 5: Add the Fractions

With both fractions having the same denominator, we can now add them. Remember, when adding fractions with a common denominator, we simply add the numerators and keep the denominator.

Our fractions are:

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

Adding the numerators, we get:

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

Now, we need to simplify the numerator by distributing and combining like terms:

6xโˆ’3+xโˆ’2(xโˆ’2)(2xโˆ’1)=7xโˆ’5(xโˆ’2)(2xโˆ’1)\frac{6x - 3 + x - 2}{(x - 2)(2x - 1)} = \frac{7x - 5}{(x - 2)(2x - 1)}

Adding the fractions is a crucial step because it brings us closer to the simplified form. However, we're not quite done yet. We need to make sure the resulting expression is in its simplest form. This means we need to check if the numerator and denominator have any common factors that can be canceled out. Combining the numerators effectively is a key part of simplifying. Ensure you distribute correctly and combine like terms accurately to avoid errors.

Step 6: Simplify the Resulting Fraction

We've added the fractions and obtained:

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

To simplify further, we need to check if the numerator, 7xโˆ’57x - 5, and the denominator, (xโˆ’2)(2xโˆ’1)(x - 2)(2x - 1), have any common factors. The numerator is a linear expression and doesn't factor further in any obvious way. The denominator is already in factored form. So, let's expand the denominator to see if it helps us identify any common factors:

(xโˆ’2)(2xโˆ’1)=2x2โˆ’xโˆ’4x+2=2x2โˆ’5x+2(x - 2)(2x - 1) = 2x^2 - x - 4x + 2 = 2x^2 - 5x + 2

Now, we have:

7xโˆ’52x2โˆ’5x+2\frac{7x - 5}{2x^2 - 5x + 2}

We need to see if 7xโˆ’57x - 5 is a factor of 2x2โˆ’5x+22x^2 - 5x + 2. Unfortunately, there's no obvious way to factor the quadratic such that (7xโˆ’5)(7x - 5) appears as a factor. We can try polynomial long division to confirm, but it's likely that there are no common factors.

Since there are no common factors, the expression is already in its simplest form.

Therefore, the simplest form of the given expression is:

7xโˆ’5(xโˆ’2)(2xโˆ’1)\frac{7x - 5}{(x - 2)(2x - 1)} or 7xโˆ’52x2โˆ’5x+2\frac{7x - 5}{2x^2 - 5x + 2}

Simplifying the resulting fraction is the final step in our process. It ensures that we've reduced the expression to its most basic form. This often involves checking for common factors and canceling them out. If we can't find any common factors, as in this case, then we know we've reached the simplest form. By reaching this stage, we've successfully navigated the simplification process, and the result is the most concise representation of the original expression. This step confirms that no further reduction is possible, giving us the final simplified algebraic expression.

Conclusion

So, guys, we've successfully simplified the given algebraic expression! We broke it down step-by-step, from factoring the denominators to finding a common denominator, adding the fractions, and finally, simplifying the result. Remember, the key to simplifying algebraic expressions is to take your time, be methodical, and double-check your work. With practice, you'll become a pro at simplifying even the most complex expressions. Keep practicing, and you'll become more confident in your algebra skills! Understanding these steps thoroughly will empower you to approach similar problems with confidence. Simplifying algebraic expressions is a fundamental skill in mathematics, and mastering it will greatly benefit you in your mathematical journey. You've got this!