Simplifying Algebraic Expressions A Step By Step Guide

Hey guys! Today, we are diving into the fascinating world of algebraic expressions. You know, those math problems that look a bit intimidating at first glance but are actually super fun once you get the hang of them. We're going to break down a specific problem step by step, so you'll not only understand the solution but also the why behind each step. Let's get started!

The Problem at Hand

So, the expression we're tackling today is:

$$\frac{x^2-7 x+10}{2 x^5} \cdot \frac{6 x^3}{3 x-15}$$

Looks a bit complicated, right? But don't worry, we'll simplify it together. Our goal is to make this expression look as clean and simple as possible. We'll do this by factoring, canceling out common terms, and using some good ol' algebraic techniques. By the end of this article, you'll be able to tackle similar problems with confidence. Trust me, it's like solving a puzzle, and the feeling when you crack it is awesome!

Step 1 Factoring Quadratics and More

The first thing we want to do is factor wherever we can. Factoring is like reverse multiplication; we're breaking down the expressions into smaller, more manageable pieces. This is crucial for simplifying algebraic expressions because it allows us to identify common factors that we can cancel out later. Think of it as organizing your tools before you start a project – it makes everything easier!

Factoring the Quadratic Expression $x^2 - 7x + 10$

Let's start with the quadratic expression in the numerator of the first fraction: $x^2 - 7x + 10$. We need to find two numbers that multiply to 10 and add up to -7. Those numbers are -2 and -5. So, we can rewrite the quadratic expression as:

$$x^2 - 7x + 10 = (x - 2)(x - 5)$$

This is a key step. Factoring the quadratic expression transforms it from a single polynomial into a product of two binomials. This makes it easier to spot common factors with other parts of the expression. Remember, practice makes perfect! The more you factor quadratic expressions, the quicker and more intuitive it will become. This skill is essential not just for simplifying expressions, but also for solving equations and understanding more advanced algebra concepts.

Factoring the Linear Expression $3x - 15$

Next, let's look at the denominator of the second fraction: $3x - 15$. Here, we can factor out a common factor of 3:

$$3x - 15 = 3(x - 5)$$

Factoring out the common factor simplifies the expression and reveals a term that we can potentially cancel out later. It’s like finding a hidden key that unlocks the simplification process. Always look for common factors in each term – it’s one of the first things you should check when simplifying algebraic expressions. This simple step can often make a big difference in making the expression more manageable.

Rewriting the Expression with Factored Forms

Now that we've factored both the quadratic expression and the linear expression, let's rewrite the original expression with these factored forms:

$$\frac{(x - 2)(x - 5)}{2 x^5} \cdot \frac{6 x^3}{3(x - 5)}$$

See how much cleaner it already looks? Factoring is like the first level in a video game – once you clear it, the path ahead becomes much clearer. We've broken down the complex parts into simpler components, and now we're ready to start canceling out those common factors. This is where the magic really happens, and the expression starts to transform into its simplified form. Keep going, guys – we're on the right track!

Step 2 Canceling Common Factors The Art of Simplification

Okay, now for the fun part – canceling out common factors! This is where we get to eliminate terms that appear in both the numerator and the denominator. It’s like decluttering your room; we're getting rid of the unnecessary stuff to make the expression cleaner and more streamlined. Remember, we can only cancel out factors that are being multiplied, not added or subtracted.

Identifying and Canceling $(x - 5)$

First, we can see that the term $(x - 5)$ appears in both the numerator and the denominator. So, we can cancel these out:

$$\frac{(x - 2)\cancel{(x - 5)}}{2 x^5} \cdot \frac{6 x^3}{3\cancel{(x - 5)}}$$

This cancellation is a big win! It simplifies the expression significantly and brings us closer to our final answer. It's like finding a matching pair in a memory game – a satisfying step towards completing the puzzle. Always be on the lookout for common factors like this; they're the key to simplifying complex expressions.

Simplifying the Constants $\frac{6}{3}$

Next, let's simplify the constants. We have $\frac{6}{3}$, which simplifies to 2. So, we can rewrite the expression as:

$$\frac{(x - 2)}{2 x^5} \cdot 2 x^3$$

Simplifying constants is a simple yet effective step. It's like tidying up the numbers before we tackle the variables. This makes the expression easier to work with and reduces the chances of making mistakes later on. Don't overlook these basic simplifications; they can make a surprisingly big difference in the overall process.

Simplifying the Variables $\frac{x^3}{x^5}$

Now, let's tackle the variables. We have $x^3$ in the numerator and $x^5$ in the denominator. When dividing variables with the same base, we subtract the exponents:

$$\frac{x^3}{x^5} = x^{3-5} = x^{-2}$$

So, we can cancel out $x^3$ from the numerator and reduce the exponent in the denominator:

$$\frac{(x - 2)}{2 x^5} \cdot 2 x^3 = \frac{(x - 2)}{2 x^2}$$

Simplifying variables using exponent rules is a fundamental skill in algebra. It's like knowing the grammar rules of a language; it allows you to manipulate and simplify expressions with precision. Remember, when you divide powers with the same base, you subtract the exponents. This rule is super handy for simplifying algebraic fractions and is something you'll use again and again in math.

Rewriting the Expression After Canceling

After canceling out the common factors and simplifying the constants and variables, we have:

$$\frac{(x - 2)}{x^2}$$

This is much simpler than our original expression, isn't it? Canceling common factors is like taking a shortcut on a long journey; it gets you to your destination much faster and with less effort. We've eliminated the unnecessary clutter and revealed the core structure of the expression. This simplified form is not only easier to look at but also easier to work with in future calculations.

Step 3 Putting It All Together The Final Simplified Form

We've reached the final step! After factoring and canceling out common factors, we have a much simpler expression. Let's put everything together and write out the final simplified form. This is the moment where all our hard work pays off, and we see the expression in its most elegant form. It's like the grand finale of a fireworks show – a satisfying conclusion to a complex process.

Combining Simplified Terms

From the previous steps, we have:

$$\frac{(x - 2)}{2 x^2} * 2$$

Notice that we still have a 2 in the numerator and a 2 in the denominator. We can cancel these out:

$$\frac{(x - 2)}{\cancel{2} x^2} \cdot \cancel{2} = \frac{x - 2}{x^2}$$

This final cancellation is like the last piece of the puzzle falling into place. It’s a small step, but it completes the simplification process and gives us our final answer. Always double-check your expression to see if there are any remaining simplifications – those little tweaks can make a big difference in the end.

The Final Simplified Expression

So, the simplified form of the original expression is:

$$\frac{x - 2}{x^2}$$

Woo-hoo! We did it! We started with a complex expression and, through factoring and canceling, we've simplified it to a much cleaner form. This simplified form is much easier to work with and understand. It's like turning a messy room into a tidy one – everything is in its place, and it's much easier to find what you need. This final simplified expression is the result of our hard work, and it's something to be proud of.

Conclusion You're an Algebra Ace!

And there you have it! We've successfully simplified a complex algebraic expression by factoring, canceling common factors, and simplifying variables. Remember, the key to simplifying algebraic expressions is to break them down step by step, look for common factors, and apply the rules of algebra. With practice, you'll become more confident and skilled at simplifying expressions. So, keep practicing, and you'll be an algebra ace in no time!

If you found this guide helpful, give yourself a pat on the back and try tackling some more problems. The more you practice, the better you'll get. And remember, math can be fun when you approach it step by step and celebrate your successes along the way. Keep up the great work, guys! You've got this!

Now, let's remember what we have learned today:

1. Factoring is the first key, it makes the expression to be simplified more easily.
2. Cancellation of common factors will remove cluttering.
3. Combining terms simplifies the equation into its final form.

With these key takeaways, you are ready to simplify more algebraic equations, Good luck!