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2.06 Graphs and characteristics of polynomial functions


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






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:


$x+3=0$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,