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.