Graphing and Finding Roots of Polynomial Functions
Remember that polynomial is just a collection of terms with coefficients and/or variables, and none have variables
in the denominator (these are Rational Expressions).
As a review, here are some polynomials, their names, and their degrees. Remember that the degree of the
polynomial is the highest exponent of one of the terms (add exponents if there are more than one variable in that
term).
Polynomial Graphs
We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or
cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative,
respectively.
Think of a polynomial graph of higher degrees (degree at least 3) as quadratic graphs, but with more twists and
turns. As a matter of fact, the maximum number of top and bottom (cup-up and cup-down) turning points
(turns) of a polynomial just happens to be one more than its degree! For example, a polynomial of degree 3,
like
The polynomial
, has 3 – 1 = 2 “turns”:
.
is in Factored Form, since we can “see all the factors”:
If we were to multiply it out, it would become
Standard Form since it’s in the form
; this is called
.
Now remember that we said that when a quadratic crosses the x axis (when y = 0), we call that point an xintercept, root, zero, solution, value, or just “solving the quadratic”. And also remember that not all of the
“solutions” were real – when the quadratic graph never touched the x axis.
Here is an example of a polynomial graph that is degree 4 and has 3 “turns” (sort of 2 “cup up”s and 1 “cup down”
in the middle). Notice that we have 3 real solutions, two of which pass through the x axis, and one “touches” it or
“bounces” off of it:
Polynomial Characteristics and Sketching Graphs
There are certain rules for sketching polynomial functions, like we had for graphing rational functions. Let’s first
talk about the characteristics we see in polynomials, and then we’ll learn how to graph them.
Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add
exponents, if you have a factored form). The leading coefficient of the polynomial is the number before the
variable that has the highest exponent (the highest degree).
So for
, the degree is 4, and the leading coefficient is 5; for
, the degree is 7 (add exponents since the polynomial isn’t multiplied out and
don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in
the factor with the highest exponent).
The end behavior of the polynomial can be determined by looking at the degree and leading coefficient. The
shape of the graphs can be determined by the x and y intercepts, end behavior, and multiplicities of each
factor. We’ll talk about end behavior and multiplicity of factors next.
End Behavior of Polynomials
The table below shows how to find the end behavior of a polynomial (which way the y is graph is “heading” as x
gets very small and x gets very large). Sorry; this is something you’ll have to memorize, but you always can figure it
out by thinking about the parent functions given in the examples:
Zeros (Roots) and Multiplicity
Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by
itself in the polynomial (its exponent). If there is no exponent for that factor, the multiplicity is 1 (which is actually
its exponent!) And remember that if you sum up all the multiplicities of the polynomial, you will get the degree!
So for example, for the factored polynomial
, the factors are x (root 0 with multiplicity 1), x –
4 (root 4 with multiplicity 2), and x + 8 (root –8 with multiplicity 3). We can ignore the leading coefficient 2, since
it doesn’t have an x in it. Also, for just plain x, it’s just like the factor x – 0. So the total of all the multiplicities of
the factors is 6, which is the degree.
Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value).
Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the
shape of the graph at that root. Factors with odd multiplicity go through the x axis, and factors with even
multiplicity bounces or touches the x axis. We are only talking about real roots; imaginary roots have similar
curve behavior, but don’t touch the x-axis.
Also note that sometimes we have to factor the polynomial to get the roots and their multiplicity. We may even
have to factor out any common factors and then do some “unfoiling” or other type of factoring (this has
a difference of squares):
Here are the multiplicity behavior rules and examples:
Now, let’s put it all together to sketch graphs; let’s find the attributes and graph the following polynomials. Notice
that when you graph the polynomials, they are sort of “self-correcting”; if you’ve done it correctly, the end
behavior and bounces will “match up”.
Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll
just want to get the basic shape.
Now you can sketch any polynomial function in factored form!
Writing Equations for Polynomials
You might have to go backwards and write an equation of a polynomial, given certain information about it:
Finding Roots (Zeros) of Polynomials
Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots. We typically do
this by factoring
For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle!
Synthetic Division
Remember that if we divide a polynomial by “x – c” and get a remainder of 0, then “x – c” is a factor of the
polynomial and “c” is a root.
Let’s use synthetic division to find all the factors and real (not imaginary) roots of the following polynomials. In
these examples, one of the factors is given, so the remainder in synthetic division should be 0. Remember to take
out a Greatest Common Factor (GFC) first, like in the second example. Notice that we can use synthetic division
again by guessing another factor, as we do in the last problem: