Rational Equations

A rational equation is an equation that can be written as

 Where N(x) and D(x) are polynomials.

 

 The Domain of a rational function includes all real numbers except for any values where D(x)=0.  this is because if D(x)=0, The function would be undefined.  However, the zeros of D(x) are still important to rational functions.  At the zeros of D(x), there is an imaginary line known as a vertical asymptote.  As the Graph approaches the asymptote, the Y values will go to infinity and negative infinity but never cross the line where D(x)=0.  

Graphs can also have horizontal asymptotes.   

If the degree of N(x) is Greater than the degree of D(x), then there is no horizontal asymptote.

If the degree of N(x) is less than the degree of D(x), then there is a horizontal asymptote at Y=0.

If the degree of N(x) is equal to the degree of D(x),then the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator.

Other points of interest include:

X-intercepts: when f(x)=0 or more simply, when N(x)=0.

Y-intercept: f(0)

Example:

Based on the equation, we can determine that there will be a vertical asymptote at x=3 and a horizontal asymptote at y=2.
By solving for f(0), we can determine that the y-intercept will be at (0, 0)
By solving for f(x)=0, we can determine that there is an x-intercept at (0, 0)

In order to describe what the graph is doing at the vertical asymptote, we must use limit notation.

Saying that the graph decreases as it moves toward the Asymptote from the left, we say

The Superscript - on the 3 indicates the left side of the graph.
Thus in order to describe the Right side of the graph we use a superscript + on the 3.

Fundamental Theory of Algebra

The fundamental theory of algebra states that any polynomial of degree n has n roots. For example, the function  has seven roots, because the highest exponent is seven. Those roots are . Five of these roots are real, but two are unreal. This can be seen in the graph:

Again, this function has five, real, visible roots and two unreal ones that cannot be easily seen on the graph. The roots we cannot see as x-intercepts can be assumed to be imaginary. This can be demonstrated in the translation of the function .

Four real roots, zero imaginary roots. 

Two reals roots, two imaginary roots.

Zero real roots, four imaginary roots.

As you probably have noticed, the roots in this function come in pairs. Real roots only come in pairs in a function that is symmetrical about the y-axis, but imaginary roots always come in pairs, no matter the shape. This is because imaginary roots come in conjugate pairs. For example, if F(2+3i)=0 then, F(2-3i)=0. 

Complex Numbers

Classification of Numbers

This is how numbers are classified:

Evaluating  

Basic power properties of i:

When i is raised to a high power:

• Divide the exponent by 4
• Find the remainder
• Take i to that power

Example:

• Divide 671 by 4 and find the remainder
• The remainder is 3
• Because , this means 

Writing Complex Numbers in Standard Form

All complex numbers (a real number added to real multiples of an imaginary unit) can be written in the standard of complex numbers:

a+bi

When a complex number is in the denominator:

• Remove it from the denominator by multiplying by its conjugate
• The conjugate is the expression formed by changing the sign between two terms

Example:

Last Thing:


There is no number between the two.
Any number added will give put you above 1.
Believe it.

Real Zeros of Polynomial Functions

Long Division of Polynomials:

We probably all remember long division right? Well, in case you dont here's an example: If we want to divide 8932 by 17 with long division what we would do is 1st see how many times 17 goes into 89, then put that number on top of the bar. Then you subtract the remainder and continue on until 17 can no longer go into the last number. That number is the remainder(in this case 7), we divide this over the original divisor and add it to the quotient. This is the quotient of 8932/17:.
Long division of Polynomials is exactly the same, but with variables for example

/x-1

Would be done like this:

This works, but takes much longer than is necessary.

Synthetic Division of Polynomials:

This is a much more simplified way to find the roots of complex polynomials called synthetic division.

It is set up like this:

Now, what number are you supposed to use as a possible zero? All the possible zeros are the factors of the last term, P, divided by the factors of the first term, Q, like so:

Calculating and plugging in these values, while still a shortcut, would still take ages. Yet again there is another shortcut. If you graph the equation, the points where it crosses the x axis, (zeros/x-intercepts) are the values which you should input, as the remainder will always be 0, meaning it is an x-intercept.

as we can see, the intercepts are, -7, -2, 3,8

 Repeat this until you have a quadratic( make sure to use the new equation, not the original), then factor the quadratic as normal. It is very possible that if the graph runs tangent to the x-axis, or exhibits some other behaviors, that there are multiples of the x-intercepts like this:

in this case the intercept at (-2,0) has a multiplicity of 3, and at (2,0) has a multiplicity of 2. This means that you must enter the -2 into synthetic division 3 times, and 3 into synthetic division 2 times

Remainder theorem:

Now lets say your teacher gives you an function with many terms. Calculating the value of 

 

would take forever. However the remainder theorem states that that if a polynomial  is divided by

the remainder is

you thought there wouldn't be scrolling text, but there was

Polynomial Functions of Higher Degree

0 Degree

1st Degree

3rd Degree

2nd Degree

4th Degree

5th Degree

Above are the graphs of varying polynomial functions. As you can see, as the degree of the function increases, the graph gets curvier. While this is not always the case, it is a general rule of thumb.

End Behavior

The ends of the graphs of the varying polynomial functions are actually pretty predictable. By looking at the leading coefficient of the term with the highest degree and the degree of the function, one can figure out how the function will look when graphed.

A positive leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach infinity. This is written as such in limit notation:

A negative leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach negative infinity. This is written as such in limit notation:

The opposite end of the graph either goes the same way as the right side or does the opposite. If the degree is EVEN, the left side acts the SAME as the right. If it is ODD, the left side acts the OPPOSITE of the right side.

Zeroes

A polynomial function has as many zeroes as its degree; however, it does not necessarily have that many X-Intercepts. A polynomial may have imaginary zeroes which are not graphed on the Cartesian plane.

Sometimes roots are repeated and can make a function that has two x intercepts only have one. This is called multiplicity.

This graph should have 4 X-Intercepts, however it only has 3. At X=0 the curve runs tangent to the X-Axis

Extrema

The graph of a polynomial function may have as many relative maxima/minima as (n-1) where n is the degree of the polynomial

Review

The graphs of polynomial functions have certain rules that allow you to predict what they will look like pretty easily. Knowing the degree and the leading coefficient allows you to make a sketch of the graph that will resemble the actual graph.

POLYNOMIAL FUNCTIONS & COMPLETEING THE SQUARE

Polynomial Functions

DEFINED AS...

A polynomial function is of the form:
- the value of  must be a nonnegative integer (meaning it is a whole number and is equal to zero or is a positive integer-- no fractions or radicals!)
- all coefficients () have to be real numbers
- the degree of the polynomial is the highest value of  where 
- is continuous
- has a domain of all real numbers
EXAMPLE

Completing the Square

Completing the square involves taking a polynomial and rewriting it in standard form.

Standard form of a quadratic is  

For EXAMPLE... 

    To rewrite this polynomial function in standard form, we will visualize the terms of the polynomial as squares.

We need one box of , 6 boxes of , and 8 boxes of 1.

When lined up together based on similar sides, attempt to fill a square.

Looks like its a square short! So we need to add 1 (but you can't just add 1, so we will ADD 1 and SUBTRACT 1 to cancel out)

Another EXAMPLE...

Send the 13 far away!

We know in order for b to have a value of -10, and the two values have to be the same, it must be:

WHICH MEANS

Since we added 50 (the 25 x 2), we must subtract 50.

Giving us the final equation of