2.3 Real Zeros of Polynomial Functions-Synthetic and Long Division Blog

In section 2.3 we look at different ways to divide polynomials. The two major ways we do this is by long division of polynomials and synthetic division.

When doing long division of polynomials you treat the problem just as you would when doing a long division problem with simple numbers

Solve the polynomial just as you would this:

Another way to divide polynomials in Synthetic Division!

This is a nice shortcut for long division. It is a pattern. Vertically you must add the terms. And your diagonal patter is to multiply by the factor outside.

In synthetic division if the integers add up to 0, (X-1) is a factor.

The Rational Zero Test
The Rational Zero Test relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term of the polynomial.
 Possible rational zeros=  (factors of constant term)/(factors of leading coefficient)

Chapter 2.2- Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

The graph of a polynomial function is:

• continuous: does not contain breaks, holes, or gaps in the graph

• has only smooth, rounded turns

Note: the graphs cannot have sharp pointed turns







Example of polynomial function:

Note: this graph is continuous- has no breaks, holes or gaps- and has smooth,

rounded turns

Monomials of the form: (Note: n must be an integer greater than zero)

n is even n is odd

When n is even, the graph touches When n is odd, the graph crosses

the axis at the x-intercept. the axis at the x-intercept.

The Leading Coefficient Test

As x moves to the left or right,

the graph rises or falls based on these rules:

When n is odd

• leading coefficient = positive, the graph falls to the left and rises to the right

• leading coefficient = negative, the graph rises to the left and falls to the right

leading coefficient is positive leading coefficient is negative



When n is even

• leading coefficient = positive, the graph rises to the left and right

• leading coefficient = negative, the graph falls to the left and right

Example

Determine the left and right behavior of the graphs:

rises to the left and falls to the right

Why? The degree is odd and the leading coefficient is negative

rises to the left and right

Why? The degree is even and the leading coefficient is positive

Zeros of Polynomial Functions

• 
  
  
  the graph f has at most n real zeros. In other words, the highest degree of a polynomial function determines the number of zeros.
  

• 
  
  the function f has at most x-1 relative extrema.

Note: Extrema are the relative minimums and maximums combined

Zero Rules:

• 
  
  
  x=a is a zero of the function f



• 
  
  
  x=a is a solution of the polynomial equation f(x)=0



• 
  
  
  (x-a) is a factor of the polynomial f(x)



• 
  
  
  (a,0) is an x-intercept of the graph f

Chapter 1.1-1.2

Piecewise Defined Functions

Piecewise defined functions are functions which symbolically define two or more formulas. For Example an absolute value graph is an example of a piecewise function.

The Function above means that any number less than zero is plugged into the top function and any number greater than or equal to zero is plugged into the bottom function.

Examples:

When these functions are graphed together the look like this:

If you have a graphing calculator you can go to the "Y=" button and type in:

After this press the graph button and you can see your graph. If you need to find a certain point on the graph you click "second" then "graph" and you find the table of points.

The Difference Quotient

The Difference Quotient is the slope of a line through the points (x, f(x)) and ( x + h, f(x + h) ).

The Difference Formula is:

----------------------

h

An example is:

-----------------------------------------------------

h

Distribute through the problem

---------------------------------------------------------

h

Cross out the terms that equal zero

--------------------

h

Factor out an h

Extra Notes:

• There should always be terms that cancel out after you distribute through the problem
• You should not, in most cases, get an answer of 1