Use your graphing calculator to graph the functions below. What are the real roots of the functions?
1.
2.
Watch This
James Sousa: Determing the Zeros or Roots of a Polynomial Function on the TI83/84
Guidance
Recall that
This means that one of the factors of the polynomial must be
Cubic polynomials are degree three and are of the form
You can use a graphing calculator to graph cubics and quartics. To graph with your graphing calculator, push [Y=], enter your polynomial, push [GRAPH] to see the graph. Then, look at the graph for information about the factors of the polynomial. You can push [TABLE] ([2nd], [GRAPH]) to see the points on the graph more clearly.
Example A
Graph the function
Solution: Once you graph the function, this is what you should see:
The polynomial
Example B
Graph the function
Solution: Once you graph the function, this is what you should see:
Since (1, 0) is an xintercept of the polynomial
Example C
How many real roots (xintercepts) are there for the polynomial
Solution: Once you graph the function, this is what you should see:
There is one xintercept so there is one real root.
Example D
Find the real root(s) for the following quartic.
Solution: This is the graph of the quartic:
There are two real roots for this quartic. One is (1, 0) and the other occurs around (2.25,0).
Concept Problem Revisited
Here is the graph of the function
Here is the graph of the function
Vocabulary
 Cubic Polynomial

A cubic polynomial is a polynomial where the largest degree is 3. So, for example,
2x3+13x2−8x+5 is a cubic polynomial.
 Real Root

A real root is a point where the graph of a function crosses the
x axis.
 Quartic Polynomials

Quartic polynomials have a degree of 4. So for example
x4−2x3−13x2−14x+24 is a quartic because it has a degree of 4.
Guided Practice
1. Find the real roots for the cubic
2. Graph the function
3. Graph the function
4. Describe the graph of the following quartic:
Answers
1.
2.
Since one of these root values is (–2, 0), the factor for the polynomial would be
3. If
4. The graph has an M shape. It looks like an M because of the –1 coefficient before
Practice
Find the real roots for the following cubic polynomials using a graph.

y=x3−2x2−9x+18 
y=x3+5x2−4x−20 
y=3x3−6x2+12x−5 
y=2x3−8x2+3x−12 
y=−2x3−3x2−5x+10
Graph the functions below and determine the number of real roots. Give at least one factor of each polynomial from the graphed solution.

y=x3−3x2−2x+6 
y=x3+x2−3x−3 
y=x3+2x2−16x−32 
y=2x3+13x2+9x+6 
y=2x3+15x2+4x−21
Graph the functions below to determine if

y=x3−2x2+3x−6 
y=x3+3x2−2x−2 
y=3x3+8x2−5x−6 
y=x3+x2−10x+8 
y=2x3−x2−3x+2
Indicate the real root(s) on the following quartic graphs:

y=x4−3x3−6x2−3 
y=x4−8x2−8  \begin{align*}y=2x^4+2x^3+x^2x8\end{align*}
 \begin{align*}y=x^46x^2x+3\end{align*}
 \begin{align*}y=x^4+x^37x^2x+6\end{align*}
Describe the following graphs:
 \begin{align*}y=x^45x^2+2x+2\end{align*}
 \begin{align*}y=x^4+3x^3x3\end{align*}
 \begin{align*}y=x^4+x^3+4x^2x+6\end{align*}
 \begin{align*}y=x^45x^35x^2+5x+6\end{align*}
 \begin{align*}y=2x^44x^35x^24x4\end{align*}