Rational Expressions

An expression that is the ratio of two polynomials:

Rational Expression

It is just like a fraction, but with polynomials.

Other Examples:

Also

re The top polynomial is "1" which is fine.
   
re Yes it is! As it could also be written:
re

But Not

no NOT the top is not a polynomial (a square root of variable is not allowed)
     
no NOT 1/x is not allowed in a polynomial

In General

A rational function is the ratio of two polynomials P(x) and Q(x) like this

Except that Q(x) cannot be zero (and anywhere that Q(x)=0 is undefined)

Finding Roots of Rational Expressions

A "root" (or "zero") is where the expression is equal to zero:

Graph of Inequality

To find the roots of a Rational Expression you only need to find the the roots of the top polynomial, so long as the Rational Expression is in "Lowest Terms".

So what does "Lowest Terms" mean?

Lowest Terms

Well, a fraction is in Lowest Terms when the top and bottom have no common factors.

Example: Fractions

2   is not in lowest terms, as 2 and 6 have the common factor "2"
6

But

1   is in lowest terms, as 1 and 3 have no common factors
3

Likewise a Rational Expression is in Lowest Terms when the top and bottom have no common factors.

Example: Rational Expressions

x3+3x2   is not in lowest terms, as x3+3x2 and 2x
have the common factor "x"
2x

But

x2+3x   is in lowest terms, as x2+3x and 2 have
no common factors
2

So, to find the roots of a rational expression:

  • Reduce the rational expression to Lowest Terms,
  • Then find the roots of the top polynomial

How do you find roots? Well you need to read Solving Polynomials to learn how!

Proper vs Improper

Fractions can be proper or improper:
Fraction Types
(There is nothing wrong with "Improper", it is just a different type)

And likewise:

A Rational Expression can also be proper or improper!

But what makes a polynomial larger or smaller?

The Degree !

For a polynomial with one variable, the Degree is the largest exponent of that variable.

Examples of Degree:

4x The Degree is 1 (a variable without an exponent actually has an exponent of 1)
   
a The Degree is 3 (largest exponent of x)


So this is how to know if a rational expression is proper or improper:

Proper: the degree of the top is less than the degree of the bottom.

Proper: 4x   4x deg(top) < deg(bottom)

Improper: the degree of the top is greater than, or equal to, the degree of the bottom.

Improper: 4x   4x deg(top) deg(bottom)

If the polynomial is improper, you might like to simplify it with Polynomial Long Division

Asymptotes

Rational expressions can have asymptotes (a line that a curve approaches as it heads towards infinity):

Example: (x2-3x)/(2x-2)

The graph of (x2-3x)/(2x-2) has:

  • A vertical asymptote at x=1
  • An oblique asymptote: y=x/2-1
   Asymptote Example

A rational expression can have:

  • any number of vertical asymptotes,
  • only zero or one horizontal asymptote,
  • only zero or one oblique (slanted) asymptote

Finding Horizontal or Oblique Asymptotes

It is fairly easy to find them ...

... but it depends on the degree of the top vs bottom polynomial.

Whichever has the larger degree will grow fastest.

Just like "Proper" and "Improper", but in fact there are four possible cases (shown below). I also put a test value "1000" for each case, just to show you what happens:

4x

Let's look at each of those examples in turn:

Degree of Top Less Than Bottom

The bottom polynomial will dominate, and there is a Horizontal Asymptote at zero.

Example: f(x) = (3x+1)/(4x2+1)

When x is 1000:

f(1000) = 3001/4000001 = 0.00075...

And for larger values of x, you will find f(x) goes very close to 0

 

Degree of Top is Equal To Bottom

Neither dominates ... the asymptote will be set by the leading terms of each polynomial.

Example: f(x) = (3x+1)/(4x+1)

When x is 1000:

f(1000) = 3001/4001 = 0.750...

And for larger values of x, you will find f(x) goes very close to 3/4

Why 3/4? Because "3" and "4" are the "leading coefficients" of each polynomial

polynomial coefficients
(The terms are in order from highest to lowest exponent)

The method is easy:

Divide the leading coefficient of the top polynomial by the leading coefficient of the bottom polynomial.

Here is another example:

Example: f(x) = (8x3 + 2x2 - 5x + 1)/(2x3 + 15x + 2)

The degrees are equal (both have a degree of 3)

Just look at the leading coefficients of each polynomial:

  • Top is 8 (from 8x3)
  • Bottom is 2 (from 2x3)

So there is a Horizontal Asymptote at 8/2 = 4

 

Degree of Top is 1 Greater Than Bottom

This is a special case: there will be an oblique asymptote, and you need to find the equation of the line.

To work it out use polynomial long division: divide the top by the bottom to find the quotient (ignore the remainder).

Example: f(x) = (3x2+1)/(4x+1)

The degree of the top is 2, and the degree of the bottom is 1, so there will ne an oblique asymptote

We need to divide 3x2+1 by 4x+1 using polynomial long division:

polynomial long division

The answer is (3/4)x-(3/16) (ignoring the remainder):

Asymptote "equation of line" is: (3/4)x-(3/16)

 

Degree of Top is More Than 1 Greater Than Bottom

When the top polynomial is more than 1 degree higher than the bottom polynomial it will just curve upwards and there is no asymptote.

Example: f(x) = (3x3+1)/(4x+1)

The degree of the top is 3, and the degree of the bottom is 1.

The top is more than 1 degree higher than the bottom so there is no horizontal or oblique asymptote.

Finding Vertical Asymptotes

There is another type of asymptote, which is caused by the bottom polynomial only.

But First: make sure the rational expression is in lowest terms!

Vertical Asymptote  

Whenever the bottom polynomial is equal to zero (ie all the roots) you get a vertical asymptote.

Read Solving polynomials to learn how to find the roots

From our example above:

Example: (x2-3x)/(2x-2)

The bottom polynomial is 2x-2, which factors into:

2(x-1)

And the factor (x-1) means there will be a vertical asymptote at x=1 (because 1-1=0)

   Asymptote Example

A Full Example

Example: Sketch (x-1)/(x2-9)

First of all, we can factor the bottom polynomial (it is the difference of two squares):

x-1
(x+3)(x-3)

Now we can see:

The roots of the top polynomial are: +1 (this is where it crosses the x-axis)

The roots of the bottom polynomial are: -3 and +3 (these will be Vertical Asymptotes)

We can also find where it crosses the y-axis which is simply where x=0:

0-1 = -1/-9 = =1/9
(0+3)(0-3)

We also know that the degree of the top is less than the degree of the bottom, so there will be a Horizontal Asymptotes at 0

So we can sketch all of that information:

 Asymptote Example

And now we can sketch in the curve:

 Asymptote Example

(You can compare that to the plot of (x-1)/(x2-9))