Simplifying Rational Expressions: A Step-by-Step Guide

by Admin 55 views
Simplifying Rational Expressions: A Step-by-Step Guide

Hey guys! Today, we're going to dive into the exciting world of simplifying rational expressions. Specifically, we'll tackle a problem that might seem a bit daunting at first, but trust me, by the end of this article, you'll be a pro at simplifying these types of quotients. Our mission, should we choose to accept it, is to simplify the following expression:

(3x^2 - 27x) / (2x^2 + 13x - 7) ÷ (3x) / (4x^2 - 1)

Sounds like a mouthful, right? But don't worry, we'll break it down step by step. Let's get started!

Step 1: Rewriting Division as Multiplication

The golden rule when dealing with division of fractions (or in this case, rational expressions) is to rewrite the division as multiplication by the reciprocal. This means we flip the second fraction and change the division sign to a multiplication sign. So, our expression becomes:

(3x^2 - 27x) / (2x^2 + 13x - 7) * (4x^2 - 1) / (3x)

See? Already it looks a little less intimidating. This is a crucial first step because it sets us up for the next phase: factoring.

Step 2: Factoring Everything

Factoring is the name of the game when simplifying rational expressions. We need to break down each polynomial into its simplest factors. This will allow us to identify common factors that can be canceled out later. Let's take each part of our expression and factor it individually.

2.1 Factoring 3x^2 - 27x

First, notice that both terms have a common factor of 3x. Let's factor that out:

3x^2 - 27x = 3x(x - 9)

Easy peasy! This is a great first step. Always look for the greatest common factor (GCF) to simplify your factoring process.

2.2 Factoring 2x^2 + 13x - 7

This is a quadratic expression, so we'll need to use our factoring skills. We're looking for two numbers that multiply to -14 (2 * -7) and add up to 13. Those numbers are 14 and -1. Let's rewrite the middle term using these numbers:

2x^2 + 13x - 7 = 2x^2 + 14x - x - 7

Now, we can factor by grouping:

2x^2 + 14x - x - 7 = 2x(x + 7) - 1(x + 7) = (2x - 1)(x + 7)

Alright, another one down! Factoring quadratics can be tricky, but practice makes perfect.

2.3 Factoring 4x^2 - 1

This expression is a difference of squares, which has a special factoring pattern: a^2 - b^2 = (a + b)(a - b). In our case, a = 2x and b = 1. So, we can factor it as:

4x^2 - 1 = (2x + 1)(2x - 1)

Sweet! Difference of squares is one of the most satisfying factoring patterns to use.

2.4 Factoring 3x

Okay, this one's pretty straightforward. 3x is already in its simplest form. We can't factor it any further. Sometimes, the simplest things are the easiest to overlook, so always give every term a quick check!

Step 3: Rewriting the Expression with Factored Forms

Now that we've factored everything, let's rewrite our entire expression using the factored forms:

[3x(x - 9)] / [(2x - 1)(x + 7)] * [(2x + 1)(2x - 1)] / [3x]

Doesn't that look like a beautiful mess? But trust me, the beauty is about to emerge as we start canceling common factors.

Step 4: Canceling Common Factors

This is the most satisfying part! Look for factors that appear in both the numerator and the denominator. We can cancel them out.

  • We have a 3x in both the numerator and the denominator, so those cancel out.
  • We also have a (2x - 1) in both the numerator and the denominator, so those cancel out too.

After canceling, our expression looks much simpler:

(x - 9) / (x + 7) * (2x + 1) / 1

Step 5: Multiplying Remaining Factors

Now, let's multiply the remaining factors in the numerator:

(x - 9)(2x + 1) = 2x^2 + x - 18x - 9 = 2x^2 - 17x - 9

Our denominator is simply (x + 7) * 1 = (x + 7).

So, our simplified expression is:

(2x^2 - 17x - 9) / (x + 7)

Step 6: State Restrictions

But we're not quite done yet! We need to state the restrictions on x. Restrictions are values of x that would make the original expression undefined, meaning any denominator would equal zero. Let's look back at our factored expressions in the denominator:

  • (2x - 1)(x + 7): This denominator gives us restrictions of x ≠ 1/2 and x ≠ -7.
  • 3x: This denominator gives us a restriction of x ≠ 0.
  • (4x^2 - 1) = (2x + 1)(2x - 1): This confirms the restriction of x ≠ 1/2. Additionally, it gives us x ≠ -1/2.

So, our restrictions are:

  • x ≠ 0
  • x ≠ 1/2
  • x ≠ -7
  • x ≠ -1/2

Final Answer

Therefore, the simplified form of the given quotient is:

(2x^2 - 17x - 9) / (x + 7), where x ≠ 0, x ≠ 1/2, x ≠ -7, and x ≠ -1/2

Key Takeaways

  • Factoring is crucial: Mastering factoring techniques is essential for simplifying rational expressions.
  • Look for common factors: Always check for the greatest common factor (GCF) first.
  • Difference of squares: Remember the pattern a^2 - b^2 = (a + b)(a - b).
  • State restrictions: Don't forget to identify and state the restrictions on x.
  • Practice makes perfect: The more you practice, the more comfortable you'll become with simplifying rational expressions.

Simplifying rational expressions might seem intimidating at first, but by breaking it down into steps and understanding the underlying principles, you can master this skill. Remember to rewrite division as multiplication, factor everything, cancel common factors, and state the restrictions. Keep practicing, and you'll be simplifying like a pro in no time!