# Multiplying and Dividing Rational Expressions

## Understanding Rational Expressions: Multiplication and Division

A rational expression is a fraction where both the numerator and denominator are polynomials. In simpler terms, it is an expression written as a ratio of two polynomials. In this article, we will cover the basics of multiplying and dividing rational expressions.

## Multiplying Rational Expressions

The process of multiplying rational expressions is similar to multiplying regular fractions. We can follow these steps to find the solution:

• Identify and cancel any common factors in the numerator and denominator by dividing both sides by the common factor. Repeat until there are no more common factors.
• Multiply the remaining numerators and denominators together.
• This is the simplified solution for the multiplication problem.

For example, let's consider the rational expression 8x²+4/16x and 3y/y. We can cancel out 4 in the numerator and denominator, giving us 2x²/4x. Simplifying this further, we get x/2 as the solution.

## Dividing Rational Expressions

The division of rational expressions follows a similar concept as multiplication. Let's take a look at the steps:

• Invert the divisor by switching the numerator and denominator. This means that the divisor becomes the new numerator and the dividend becomes the new denominator.
• Proceed with the same steps as multiplying rational expressions.

For instance, if we have 2x/4 divided by 1/3, we can invert 1/3 to get 3/1 and follow the steps for multiplication. This gives us 6x as the solution.

## Examples of Multiplying and Dividing Rational Expressions

Let's see how we can use these steps to solve some examples:

### Example 1:

(2x+4)/(x²+3x+2) x (x²+2x+1)/(8x+16)

First, we can factorize the first rational expression to (2x+4)/(x+2)(x+1) and the second one to (x+1)/(8(x+2)). By cancelling out the common factor (x+1) in both expressions, we get 2/(8x+16). This can be further simplified to 1/4x+4.

### Example 2:

(3x+9)/(x²+3x+2) ÷ (x²+3x+2)/(6x+12)

Here, we invert the second rational expression to get (6x+12)/(x²+3x+2). Simplifying the first expression to (3x+9)/(x+1)(x+2), we can cancel out common factors and get 3/(1)(x+2), which can be simplified to 3/(x+2). Following the steps for multiplication, we get 6/(6x+12) and simplifying it gives us 1/(x+2) as the solution.

## Key Takeaways

• A rational expression is a fraction where both the numerator and denominator are polynomials.
• To multiply rational expressions, we multiply the numerators and denominators together.
• To divide rational expressions, we invert the divisor and follow the same steps as multiplication.
• Standard polynomial factorization methods can be applied to factorize the numerators and denominators in rational expressions.