2.6 Rational Functions and Asymptotes

2.6 Rational Functions and Asymptotes

A rational function is a function that can be written:

N(x) and D(x) are both polynomials.

A ratio of two rational functions is rational.

The domain of the function is all real numbers except the values that make D(x)= 0. D(x) cannot equal zero because that would make the function undefined.

The vertical asymptote is an imaginary line that a graphed function gets really close to (the y values go to infinity and negative infinity) but never touch this "line". This is because at this x value, the function is undefined.

  as  (from the right)

 as  (from the left)

The horizontal asymptote is an imaginary line line that a graphed function gets really close to (the x values go to infinity and negative infinity) but never touch this "line". This is because at this y value, the function is undefined.

 as 

 as 

To find the x-intercept(s), set:

aka:

To find the y-intercept, set:

To find the vertical asymptote(s), set:

• When finding the vertical asymptote, you can disregard anything after the first term, as it is in significant. You can imagine this by plugging in infinity for x. 2(Infinity) over  just infinity is much more significant than infinity over infinity.
• Example:

so 3 is the vertical asymptote.

To find the horizontal asymptote(s):

• If the degree of x on the numerator is less than the degree of x on the denominator, the horizontal asymptote is 0. If imagine x= , then ² is a lot bigger than just .
• If the degree of x on the numerator is equal to the degree of x on the denominator, the horizontal asymptote is just the numerator's leading coefficient over the denominator's leading coefficient.
• If the degree of x on the numerator is equal to the degree of x on the denominator, there is no horizontal asymptote.
• Example:

so 2 is the horizontal asymptote.

Graph of :

x-intercept: none

y-intercept: none

vertical asymptote: 0

horizontal asymptote: 0

Graph of:

x-intercept: 0

y-intercept: 0

vertical asymptote: -3

horizontal asymptote: 2

picture from: http://talkrational.org/showthread.php?t=37586

2.5 The Fundamental Theorem of Algebra 

The fundamental theorem of algebra states that a polynomial function of a degree n has exactly n zeroes. This is true if n>0. These zeroes can either be real or non-real numbers.

So, whether real or non-real, the function  has exactly 2 zeroes, the function  has exactly 3 zeroes, and so on.


Real Zeroes of Polynomial Functions

• Some functions have zeroes that are all real 

Consider the function:


when factored, the function simplifies to

(x+4)(x+2)

so, the resulting zeroes of this function are -4 and -2. This function has exactly 2 real zeroes. 

Non-Real Zeroes of Polynomial Functions

Although it is nice when when all of the zeroes of a function are real, this is often not the case. Many functions have non-real zeroes. 

• Non-real zeroes are written in the form a+bi
• Non-real zeroes ALWAYS COME IN PAIRS!!! This means that any given function can only have an EVEN number of non-real zeroes 
• The pairs that non-real zeroes come in are the non-real zero and its conjugate

A function  has 4 real zeroes 

when factored, it comes out to be: 
     
           

NOTICE: The non real zero  and its conjugate  - are both zeroes

Graphing Polynomial Functions with Non-Real Zeroes 

From this illustration, it is demonstrated that non-real zeros are zeros that DO NOT touch or cross the x axis on the graph of a function. 

In the graph of the function of degree 4 with no real zeros, it does not touch or cross the x axis. This means that the function still does have exactly 4 zeros, but they are all non-real

On the graph of the function with the degree of four with four real zeros, it crosses the x axis 4 times, at each point the graph crosses the x axis, this a real zero of the function

On the graph of the function with a degree of 6, it is demonstrated that this function has both real and non-real zeros. The points where x changes direction but does not touch the x axis represent the non-real zeros of the function. The two points where the graph crosses the x axis represent the real zeros of the function.

TRUE or FALSE

An equation with zero  must also have a zero at . TRUE 

• This statement is true because non-real zeros of a polynomial function always come in pairs 

It is possible for the equation  to only have 3 zeros. FALSE

• This statement is false because the fundamental theorem of algebra states that a polynomial to the nth degree must have exactly n zeros

2.3 Real Zeros of Polynomial Functions

2.3 Real Zeros of Polynomial Functions

Polynomials can be factored to find their x-intercepts (also known as their zeros). However, polynomials that are raised to high degrees can be difficult to factor. This is when long division and synthetic division can be helpful!

Long Division of Polynomials

Long division is a basic math tool that is very helpful when dealing with advanced math. For example, it can be used to divide 

by x-2.

Heres how:

Its just like normal long division. Say to yourself...

1. x (from x-2) goes into x cubed how many times? Well x squared times!! 

2. Now multiply the x squared by the x-2 and subtract that from the x cubed minus 9 x squared.

3. After that is done the x cubed term should cancel out and you should be left with -7x squared. Now bring down the next term (26x)

4. Repeat! How many times does x go into  -7x squared? -7x times! And continue until you have brought down all possible terms.

If you are left with 0, then great! Whatever you have written as your quotient is your final answer. BUT, if you are left with anything other than 0, you have a remainder. So what does that mean?

Take a look...

In this case, the remainder is -1. Instead of simply writing the quotient as your final answer, you have to remember to also add the remainder divided by the dividend. This would look like:

Another important trick for long division of polynomials is to leave space for x raised to degrees that may not be present in your dividend. For example, if the dividend is x cubed minus 1 you would write:

Notice the amount of space left!! This way, when writing the quotient and doing work for long division, all terms remain lined up properly.

Synthetic Division 

Synthetic division is a shortcut for factoring really long, complicated polynomials. Here's how it works:

Lets say you want to factor f(x)=

.

First you set up your synthetic division. 

Take each coefficient and line them up in order like so. Notice that if a term is negative, the coefficient is negative when written for synthetic division. In this example, we will be trying to see if 2 is a factor, so the 2 is written off to the side. 

The first step of synthetic division is simple. You just take the first coefficient, and bring it straight down.

Now, you take the number under the line and multiply it by the number to the left of the line. So, in this case 2*2=4.

Next, just add straight down. So, since 7+4=11, write 11 under the line. You should continue this pattern of multiplying then adding until your synthetic division looks like this:

In this case, the remainder is 0. This means that 2 is indeed a factor of this polynomial. If the number in the red box were not 0, then 2 would not be a factor of 0. 

After doing synthetic division, you are left with some numbers written under a line. Are they helpful? Turns out, they are exactly what you were looking for. These are the coefficients for the newly factored function. The newest factored form of the equation is...

To fully factor this equation, simply factor repeat synthetic division for the remaining polynomial and then factor by hand until you cannot factor it any more.

At this point a common question comes to mind... How do you know which number to try to factor with? Do you really just have to pick from an infinite amount of numbers until one magically works?

Well don't worry, there's a simple way to narrow down the list. 

To make a list of possible factors simply write: 

factors of constant / factors of coefficients of term to highest degree

So, for the synthetic division done earlier in this blog, that list would be...

It may still look like a lot of options for a polynomial that will have at most 4 real solutions, but it is still a lot less than an infinite number of options.