What Is The Difference?A. $\frac{2x-3}{3x^2}-\frac{x+2}{6x}$B. $\frac{x-5}{6x^2}$C. $\frac{3x-8}{6x^2}$D. $\frac{-x^2+6x-6}{6x^2}$E. $\frac{-x^2+2x-6}{6x^2}$

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What is the Difference? Simplifying Rational Expressions

In mathematics, rational expressions are a crucial part of algebraic manipulations. They are formed by the ratio of two polynomials and can be simplified using various techniques. In this article, we will explore the difference between five given rational expressions and simplify them to their lowest terms.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and constants in the numerator and denominator. It can be written in the form of p(x)/q(x), where p(x) and q(x) are polynomials. Rational expressions can be simplified by canceling out common factors in the numerator and denominator.

Simplifying Rational Expressions

To simplify a rational expression, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest factor that divides both the numerator and denominator without leaving a remainder. Once we find the GCF, we can cancel it out from both the numerator and denominator.

Simplifying Option A

Let's start by simplifying option A: 2x−33x2−x+26x\frac{2x-3}{3x^2}-\frac{x+2}{6x}. To simplify this expression, we need to find the GCF of the numerator and denominator of each fraction.

# Simplifying Option A
## Step 1: Find the GCF of the numerator and denominator of each fraction
The GCF of 2x-3 and 3x^2 is x-3.
The GCF of x+2 and 6x is x.

## Step 2: Cancel out the GCF from both the numerator and denominator
$\frac{2x-3}{3x^2}-\frac{x+2}{6x} = \frac{(x-3)(2x-3)}{x(3x^2)} - \frac{x(x+2)}{6x^2}$

Simplifying Option B

Now, let's simplify option B: x−56x2\frac{x-5}{6x^2}. To simplify this expression, we need to find the GCF of the numerator and denominator.

# Simplifying Option B
## Step 1: Find the GCF of the numerator and denominator
The GCF of x-5 and 6x^2 is 1.

## Step 2: Cancel out the GCF from both the numerator and denominator
$\frac{x-5}{6x^2} = \frac{x-5}{6x^2}$

Simplifying Option C

Next, let's simplify option C: 3x−86x2\frac{3x-8}{6x^2}. To simplify this expression, we need to find the GCF of the numerator and denominator.

# Simplifying Option C
## Step 1: Find the GCF of the numerator and denominator
The GCF of 3x-8 and 6x^2 is 2x-4.

## Step 2: Cancel out the GCF from both the numerator and denominator
$\frac{3x-8}{6x^2} = \frac{(2x-4)(3x-2)}{2x(3x^2)}$

Simplifying Option D

Now, let's simplify option D: −x2+6x−66x2\frac{-x^2+6x-6}{6x^2}. To simplify this expression, we need to find the GCF of the numerator and denominator.

# Simplifying Option D
## Step 1: Find the GCF of the numerator and denominator
The GCF of -x^2+6x-6 and 6x^2 is -x+3.

## Step 2: Cancel out the GCF from both the numerator and denominator
$\frac{-x^2+6x-6}{6x^2} = \frac{-(x-3)(x-2)}{6x^2}$

Simplifying Option E

Finally, let's simplify option E: −x2+2x−66x2\frac{-x^2+2x-6}{6x^2}. To simplify this expression, we need to find the GCF of the numerator and denominator.

# Simplifying Option E
## Step 1: Find the GCF of the numerator and denominator
The GCF of -x^2+2x-6 and 6x^2 is -x+3.

## Step 2: Cancel out the GCF from both the numerator and denominator
$\frac{-x^2+2x-6}{6x^2} = \frac{-(x-3)(x-2)}{6x^2}$

Conclusion

In this article, we simplified five rational expressions by finding the greatest common factor (GCF) of the numerator and denominator and canceling it out. We also compared the simplified expressions to determine which one is different from the others. The simplified expressions are:

  • Option A: (x−3)(2x−3)x(3x2)−x(x+2)6x2\frac{(x-3)(2x-3)}{x(3x^2)} - \frac{x(x+2)}{6x^2}
  • Option B: x−56x2\frac{x-5}{6x^2}
  • Option C: (2x−4)(3x−2)2x(3x2)\frac{(2x-4)(3x-2)}{2x(3x^2)}
  • Option D: −(x−3)(x−2)6x2\frac{-(x-3)(x-2)}{6x^2}
  • Option E: −(x−3)(x−2)6x2\frac{-(x-3)(x-2)}{6x^2}

The difference between the simplified expressions is that option A has a subtraction sign between the two fractions, while the other options do not.
Frequently Asked Questions (FAQs) About Simplifying Rational Expressions

In our previous article, we explored the difference between five given rational expressions and simplified them to their lowest terms. In this article, we will answer some frequently asked questions (FAQs) about simplifying rational expressions.

Q: What is the greatest common factor (GCF) and why is it important in simplifying rational expressions?

A: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator of a rational expression without leaving a remainder. It is important in simplifying rational expressions because it allows us to cancel out common factors and simplify the expression.

Q: How do I find the GCF of two polynomials?

A: To find the GCF of two polynomials, you can use the following steps:

  1. Factor both polynomials into their prime factors.
  2. Identify the common factors between the two polynomials.
  3. Multiply the common factors together to find the GCF.

Q: What is the difference between a rational expression and a rational number?

A: A rational expression is a fraction that contains variables and constants in the numerator and denominator, while a rational number is a fraction that contains only integers in the numerator and denominator.

Q: Can I simplify a rational expression if the numerator and denominator have no common factors?

A: Yes, you can simplify a rational expression even if the numerator and denominator have no common factors. In this case, the expression is already in its simplest form.

Q: How do I simplify a rational expression with a negative sign in the numerator or denominator?

A: To simplify a rational expression with a negative sign in the numerator or denominator, you can follow the same steps as before. However, be careful when canceling out common factors, as the negative sign may affect the result.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you must be careful when canceling out common factors, as the variable may affect the result.

Q: What is the final answer to the problem of simplifying the five given rational expressions?

A: The final answer to the problem of simplifying the five given rational expressions is:

  • Option A: (x−3)(2x−3)x(3x2)−x(x+2)6x2\frac{(x-3)(2x-3)}{x(3x^2)} - \frac{x(x+2)}{6x^2}
  • Option B: x−56x2\frac{x-5}{6x^2}
  • Option C: (2x−4)(3x−2)2x(3x2)\frac{(2x-4)(3x-2)}{2x(3x^2)}
  • Option D: −(x−3)(x−2)6x2\frac{-(x-3)(x-2)}{6x^2}
  • Option E: −(x−3)(x−2)6x2\frac{-(x-3)(x-2)}{6x^2}

The difference between the simplified expressions is that option A has a subtraction sign between the two fractions, while the other options do not.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about simplifying rational expressions. We covered topics such as the greatest common factor (GCF), finding the GCF of two polynomials, and simplifying rational expressions with negative signs and variables in the denominator. We also provided the final answer to the problem of simplifying the five given rational expressions.