Simplifying Algebraic Expressions: A Step-by-Step Guide

Oct 22, 2025 by ADMIN 56 views

Hey math enthusiasts! Let's dive into the fascinating world of algebraic expressions. Today, we're going to tackle two problems that involve simplifying expressions, including division and subtraction of rational expressions. Don't worry, it's not as scary as it sounds! We'll break it down step by step, so you can follow along easily. By the end of this guide, you'll be a pro at simplifying these types of expressions. Ready to get started? Let's go!

Finding the Simplified Quotient: Division of Rational Expressions

Alright, guys, our first challenge is to simplify the quotient of two rational expressions: $\frac{t^2-36}{3} \div \frac{t^2+6 t}{9 t}$ . This might look a bit intimidating at first, but trust me, it's manageable. Remember, when dividing fractions, we actually multiply by the reciprocal of the second fraction. Let's start by rewriting the problem using this principle.

First, we'll flip the second fraction and change the division to multiplication. That gives us:

$\frac{t^2-36}{3} \times \frac{9 t}{t^2+6 t}$

Now, before we go multiplying everything out, let's factor as much as possible. Factoring is your best friend when simplifying algebraic expressions! The numerator of the first fraction, $t^2 - 36$ , is a difference of squares. This means it can be factored into $(t - 6)(t + 6)$ . The second fraction's numerator is $9t$ , which is already in its simplest form. For the denominator of the second fraction, $t^2 + 6t$ , we can factor out a $t$ , which gives us $t(t + 6)$ .

So, our expression now looks like this:

$\frac{(t - 6)(t + 6)}{3} \times \frac{9 t}{t(t + 6)}$

See how much simpler it’s already looking? Next, we can cancel out common factors that appear in the numerator and denominator. We can cancel out the $(t + 6)$ from the numerator and the denominator, and also simplify the numbers. We can divide $9$ by $3$ to get $3$ . This leaves us with:

$\frac{(t - 6)}{1} \times \frac{3}{1}$

Multiply across and we get:

$3(t - 6)$

Or,

$3t - 18$

And there you have it! The simplified form of the given expression is $3t - 18$ . See? Not so bad, right?

Breaking Down the Simplification Process

Let's recap the steps we took to simplify this expression. We began with division, converted it to multiplication by using the reciprocal, factored the expressions, cancelled out common factors, and simplified the result. Each step is crucial, and the order matters. Always remember to factor first, simplify the numbers, and then cancel out any common factors. The key takeaway here is to always look for opportunities to factor and simplify before multiplying everything out. This will make your life much easier!

Simplifying the Expression: Subtraction of Rational Expressions

Now, let's move on to our second problem. This time, we're going to simplify a complex expression involving subtraction: $\frac{2 x^2+5 x+3}{x^2-3 x-4}-\frac{4 x^2+2 x-6}{x^2-8 x+16}$ . This one requires a few more steps, but we've got this! The key here is to find a common denominator and then combine like terms. This process is similar to adding or subtracting regular fractions.

First, let's factor the numerators and denominators wherever possible. For the first fraction, $2x^2 + 5x + 3$ can be factored into $(2x + 3)(x + 1)$ , and $x^2 - 3x - 4$ can be factored into $(x - 4)(x + 1)$ . For the second fraction, $4x^2 + 2x - 6$ can be factored into $2(2x^2 + x - 3)$ , which then breaks down to $2(2x + 3)(x - 1)$ , and $x^2 - 8x + 16$ is a perfect square trinomial, factoring into $(x - 4)^2$ or $(x - 4)(x - 4)$ .

So, our expression now looks like this:

$\frac{(2x + 3)(x + 1)}{(x - 4)(x + 1)} - \frac{2(2x + 3)(x - 1)}{(x - 4)(x - 4)}$

Next, we need to find a common denominator. Looking at the denominators, we have $(x - 4)(x + 1)$ and $(x - 4)(x - 4)$ . The least common denominator (LCD) will be $(x - 4)(x - 4)(x + 1)$ .

To get each fraction to have this LCD, we need to multiply the first fraction by $\frac{(x - 4)}{(x - 4)}$ . The second fraction already has a factor of $(x-4)$ so it needs to be multiplied by $\frac{(x + 1)}{(x + 1)}$ .

This gives us:

$\frac{(2x + 3)(x + 1)(x - 4)}{(x - 4)(x + 1)(x - 4)} - \frac{2(2x + 3)(x - 1)(x + 1)}{(x - 4)(x - 4)(x + 1)}$

Combining and Simplifying the Expression

Now that we have a common denominator, we can combine the numerators. This step involves some careful distribution and combining like terms. Expanding the numerator for the first fraction we have $(2x^2 - 8x + 3x - 12)$ , which becomes $(2x^2 - 5x - 12)$ . Expanding the numerator for the second fraction we get $2(2x^2 + 2x - 3x - 3)$ , which becomes $2(2x^2 - x - 3)$ , which simplifies to $(4x^2 - 2x - 6)$ . Now we can subtract the second numerator from the first one.

So, our expression becomes:

$\frac{(2x + 3)(x + 1)(x - 4) - 2(2x + 3)(x - 1)(x + 1)}{(x - 4)(x + 1)(x - 4)}$

Combine the numerators:

$\frac{(2x^2 - 5x - 12) - (4x^2 - 2x - 6)}{(x - 4)(x - 4)(x + 1)}$

Now subtract the second numerator from the first:

$\frac{2x^2 - 5x - 12 - 4x^2 + 2x + 6}{(x - 4)^2(x + 1)}$

Combine like terms:

$\frac{-2x^2 - 3x - 6}{(x - 4)^2(x + 1)}$

At this point, you can look for common factors to simplify the expression further, but in this case, the numerator and denominator don't share any common factors. Therefore, this is the simplified form of the expression.

Mastering Rational Expressions

This problem was a bit more involved, but you did it! Remember, the key steps were factoring, finding a common denominator, combining numerators, and simplifying. Take your time with each step, and don’t rush. Practice makes perfect, so keep practicing these types of problems, and you'll become a whiz at simplifying rational expressions. Keep in mind that understanding factoring is absolutely critical for these types of problems.

Tips for Success in Simplifying Expressions

Alright guys, before we wrap up, let's go over a few essential tips to help you succeed in simplifying algebraic expressions. These tips will not only help you solve these types of problems but will also help you in other areas of mathematics as well.

Factor, Factor, Factor

Embrace Factoring: Factoring is your most important tool. Become familiar with different factoring techniques, such as factoring out the greatest common factor (GCF), factoring by grouping, and recognizing special products (like the difference of squares and perfect square trinomials). The more comfortable you are with factoring, the easier it will be to simplify expressions.

Common Denominators

Find Common Denominators Carefully: When adding or subtracting fractions, always find the least common denominator (LCD). This is the smallest expression that all denominators divide into evenly. The LCD simplifies the process and reduces the chances of errors.

Combine Like Terms

Combine Like Terms Methodically: After finding the common denominator and combining numerators, carefully combine like terms. This involves adding or subtracting terms with the same variable and exponent. Be meticulous, and double-check your work to avoid making mistakes.

Simplify, Simplify, Simplify

Always Simplify: After performing all the operations, simplify the expression as much as possible. Cancel out common factors in the numerator and denominator. Your final answer should always be in its simplest form.

Practice Makes Perfect

Practice Regularly: The best way to get better at simplifying algebraic expressions is through practice. Work through different types of problems, and don't be afraid to make mistakes. Each mistake is an opportunity to learn and improve.

Check your Work

Check Your Work: It's always a good idea to check your work, especially when dealing with complex expressions. You can do this by plugging in a value for the variable and seeing if the original expression and the simplified expression yield the same result.

Conclusion

And there you have it, guys! We've successfully simplified two algebraic expressions. Remember, practice is key. Keep working at it, and you’ll master these skills in no time. With a little practice and patience, you'll be simplifying algebraic expressions like a pro. Keep up the great work, and happy calculating!