# 2.06 Graphs and characteristics of polynomial functions

Lesson

If we start with the most basic form of a polynomial, $y=ax^n$y=axn, we can immediately see a pattern emerge for odd and even powers. Experiment with the following applet, where $a=1$a=1.

Odd degree polynomials move in opposite directions at the extremities and even degree polynomials move in the same direction at the extremities! This is because the degree of a polynomial, $n$n, together with the leading coefficient, $a$a, dictate the overall shape and behavior of the function at the extremities:

$n$n $a>0$a>0 $a<0$a<0

Even

Odd

But what happens between the extremities?

A polynomial of degree $n$n can have up to $n$n $x$x-intercepts, with those of odd degree having at least one.

A polynomial of degree $n$n can have up to $n-1$n1 turning points, with those of even degree having at least one.

Details of key features can be found using a table of values and technology or from the factored forms of the function.  Let's focus on what we can learn from factored forms of polynomials.

### Factored polynomials and multiple roots

Recall the zero product property from when we solved quadratic equations previously:

Zero product property

If $ab=0$ab=0, then $a=0$a=0 or$b=0$b=0

For example:

$\left(x+3\right)\left(x-4\right)=0$(x+3)(x4)=0

$x+3=0$x+3=0 or $x-4=0$x4=0

$x=-3$x=3 or $x=4$x=4

We will extend this concept to polynomials of any degree, so if we can fully factor a polynomial, then we can quickly find the $x$x-intercepts, solutions or roots.

Certain polynomials of degree $n$n can be factored into up to $n$n linear factors over the real number field. For example the $4$4th degree polynomial $P\left(x\right)=2x^4-x^3-17x^2+16x+12$P(x)=2x4x317x2+16x+12 can be expressed as $P\left(x\right)=\left(x+3\right)\left(2x+1\right)\left(x-2\right)^2$P(x)=(x+3)(2x+1)(x2)2. Note that there are two distinct factors, the $\left(x+3\right)$(x+3) and $\left(2x+1\right)$(2x+1) and two equal factors, the $\left(x-2\right)$(x2) appears twice. But we can immediately identify that there are roots at $x=-3,-\frac{1}{2}$x=3,