# Polynomial Functions

$f(x) = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$

by changing the values of the coefficients $a, b, c, d, e$ and $f$. It is not easy to draw any conclusion when you change all 5 coefficients at the same time. You can always reduce the degree (highest power) by setting some parameters to zero. For example if you set coefficients $a$ to zero and $b$ to a non zero value, you obtain a polynomial of degree 4. Once you finish this interactive tutorial, you may want to consider a Graphs of polynomial functions - Questions. If needed, Free graph paper is available.

## Interactive Tutorials Using an App

A polynomial f(x) with real coefficients and of degree n has n zeros (not necessarily all different). Some or all are real zeros and appear as x-intercepts when f(x) is graphed.

## A - Explore Real Solutions of Polynomial Equations of the Form

$x^n + f = 0$

where $n$ is even or odd and $f$ is a constant.

 $a$ = 1 $b$ = 2 $c$ = -2 $d$ = -3 $e$ = 1 $f$ = -2

Click the button "Plot Polynomial" to start.

1. Set all coefficients to zero except $a$ and $f$.
Write down the polynomial and its degree, examine the graph obtained. How many x-intercepts ( or real solutions to the above equation ) the graph has? Repeat for different values of a and f

2. Set all coefficients to zero except $b$ and $f$.
Write down the polynomial and its degree, examine the graph obtained. Change b and f and see how many x-intercepts the graph has? Which values of f give intercepts and which values do not give any intercepts?

3. Set all coefficients to zero except c and f.
Write down the polynomial and its degree, examine the graph you obtain. Change c and f and see how many x-intercepts the graph has?

4. Set all coefficients to zero except d an f.
Write down the polynomial and its degree, examine the graph you obtain. Change d and c and see how many x-intercepts the graph has and for what values of f

5. Set all coefficients to zero except e and f,
write down the polynomial and its degree, examine the graph you obtain. Change e and f and see how many x-intercepts the graph has?

From 1,2,3,4, and 5 above, what conclusion can you make as to the number of solutions of polynomials equations of the form $x^n + f = 0$
depending on whether $n$ is even or odd and $f$ is negative, positive or zero?.

## B - Explore Even and Odd Polynomials

1. Set a, c and e to zero, write down the
polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

2. Set b,d and f to zero, write down the
polynomial and its degree, examine the graph you obtain, is f(x) even, odd or neither?

## C - Zeros of Polynomials

NOTE: For the next tutorial, a repeated zero of multiplicity m is counted m times.

1. Set $a$ to a non zero value (polynomial of degree 5). Change all the other coefficients (non zero values if possible) so that the graph of $f(x)$ has:
1 x-intercept.
3 x-intercepts.
5 x-intercepts.
Why do you think we can obtain only an odd number of real zeros of $f(x)$?

2. Set $a$ to zero and $b$ to a non zero value (polynomial of degree 4). Change all the other coefficients (non zero values if possible) so that the graph of $f(x)$ has:
no x-intercept.
2 x-intercepts.
4 x-intercepts.
Why do you think we can obtain only an even number of real zeros for $f(x)$?

(The last two exercises are not easy, however they are very useful to fully understand polynomials).

## D - Leading Coefficient Test

1. Set the leading coefficient $a$ to a positive value (polynomial of degree 5) and set b, c, d, e and f to some values.
As x increases without bounds, does the right side of the graph rise or fall?
As x decreases without bounds, does the right side of the graph rise or fall?
Change $b, c, d, e$ and $f$ and see if the above behavior changes.

2. Set $a$ to zero and $b$ (leading coefficient) to a positive value to obatin a polynomial of degree 4 and carry out the same exploration as in 1 above and 2 above.

3. Set a and b to zero and c (leading coefficient) to a positive value (polynomial of degree 3) and do the same exploration as in 1 above and 2 above.

4. Set a, b and c to zero and d (leading coefficient) to a positive value (polynomial of degree 2) and do the same exploration as in 1 above and 2 above.

5. Set a, b, c and d to zero and e (leading coefficient) to a positive value (polynomial of degree 1) and do the same exploration as in 1 above and 2 above.

What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

What can you say about the behavior of the graph of the polynomial f(x) with an even degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a positive leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

What can you say about the behavior of the graph of the polynomial f(x) with a odd degree n and a negative leading coefficient as x increases without bounds? What do you say about the behavior of the same polynomial as x decreases without bounds?

## More references and links to polynomial functions

Derivatives of Polynomial Functions.
Polynomial Functions, Zeros, Factors and Intercepts
Find Zeros of Polynomial Functions - Problems
Multiplicity of Zeros and Graphs Polynomials.
Graphs of Polynomial Functions - Questions.