6.4 - Rational Expressions

6.4 - Rational Expressions

Rational Expression

A quotient of two polynomials

Examples

•  $\frac{5}{x}$  

•  $\frac{x^2-4}{x+2}$  

•  $\frac{x+3}{x-7}$  

Because rational expressions are ratios of polynomials, you can simplify them the same way you simplify fractions

•  $\frac{9}{24}=\frac{3\cdot3}{8\cdot3}=\frac{3}{8}$  

• Divide out common factors in the numerator and denominator

Example 1

•  $\frac{x^2-2x-3}{x^2+5x+4}$  

•  $\frac{\left(x-3\right)\left(x+1\right)}{\left(x+1\right)\left(x+4\right)}$  

•  $\frac{\left(x-3\right)}{\left(x+4\right)}$  

• STOP!

We can't simplify $\frac{x-3}{x+4}$  any further.

• Specifically, we cannot divide out the x terms.

• That would be like saying I'm going to simplify  $\frac{66}{63}$  by dividing out the 6s and saying it's equal to  $\frac{6}{3}=2$  

• It doesn't work that way

One other thing

• We have to check for values of x that restricted from the domain.

• In  $\frac{\left(x-3\right)\left(x+1\right)}{\left(x+1\right)\left(x+4\right)}$  , the expression is undefined at x=-1 and x=-4 because those values of x make the factors (x+1) and (x+4) equal to 0, making the whole denominator equal to 0. And division by 0 is undefined.

You try

Multiple Choice

Simplify $\frac{x^2+8x+15}{x^2+x-6}$  

•  $\frac{8x+15}{x-6}$  

•  $\frac{x+5}{x-2}$  

•  $7x+9$  

•  $\frac{x+3}{x-2}$  

We can also multiply and divide rational expressions

You try

Multiple Choice

Multiply $\frac{10x-40}{x^2-6x+8}\cdot\frac{x+3}{5x+15}$  

•  $\frac{10}{5}$  

•  $5x-10$  

•  $\frac{2}{x-2}$  

•  $\frac{1}{x}$  

We'll get a chance to practice dividing rational expressions in the Math XL set after the break