Use your graphing calculator to graph the functions below. What are the real roots of the functions?
1. \begin{align*}f(x)=x^36x^2+11x6\end{align*}
2. \begin{align*}g(x)=2x^44x^33x^2+12x8\end{align*}
Watch This
James Sousa: Determing the Zeros or Roots of a Polynomial Function on the TI83/84
Guidance
Recall that \begin{align*}xa\end{align*}
This means that one of the factors of the polynomial must be \begin{align*}(x+3)\end{align*}
Cubic polynomials are degree three and are of the form \begin{align*}y=ax^3+bx^2+cx+d\end{align*}
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 \begin{align*}f(x)=x^3+x^2+x3\end{align*}
Solution: Once you graph the function, this is what you should see:
The polynomial \begin{align*}f(x)=x^3+x^2+x3\end{align*}
Example B
Graph the function \begin{align*}g(x)=x^35x^2+8x4\end{align*}
Solution: Once you graph the function, this is what you should see:
Since (1, 0) is an xintercept of the polynomial \begin{align*}g(x)=x^35x^2+8x4\end{align*}
Example C
How many real roots (xintercepts) are there for the polynomial \begin{align*}h(x)=x^35x^22x+24\end{align*}
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.
\begin{align*}k(x)=x^43x^3+2x^2x+1\end{align*}
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 \begin{align*}f(x)=x^36x^2+11x6\end{align*}
Here is the graph of the function \begin{align*}g(x)=2x^44x^33x^2+12x8\end{align*}
Vocabulary
 Cubic Polynomial

A cubic polynomial is a polynomial where the largest degree is 3. So, for example, \begin{align*}2x^3+13x^28x+5\end{align*}
2x3+13x2−8x+5 is a cubic polynomial.
 Real Root

A real root is a point where the graph of a function crosses the \begin{align*}x\end{align*}
x axis.
 Quartic Polynomials

Quartic polynomials have a degree of 4. So for example \begin{align*}x^42x^313x^214x+24\end{align*}
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 \begin{align*}y=x^3+3x4\end{align*}
2. Graph the function \begin{align*}g(x)=3x^3+8x^2+3x2\end{align*}
3. Graph the function \begin{align*}m(x)=2x^3+10x^2+8x1\end{align*}
4. Describe the graph of the following quartic: \begin{align*}j(x)=x^43x^3+2x^2+x6\end{align*}.
Answers
1.
2.
Since one of these root values is (–2, 0), the factor for the polynomial would be \begin{align*}(x + 2)\end{align*} and not \begin{align*}(x  2)\end{align*}.
3. If \begin{align*}(x  1)\end{align*} was one of the factors then one of the roots would have to be (1, 0). This is not the case.
4. The graph has an M shape. It looks like an M because of the –1 coefficient before \begin{align*}x^4\end{align*}. There are two real roots located at (–3.35, 0) and (–1.32, 0).
Practice
Find the real roots for the following cubic polynomials using a graph.
 \begin{align*}y=x^32x^29x+18\end{align*}
 \begin{align*}y=x^3+5x^24x20\end{align*}
 \begin{align*}y=3x^36x^2+12x5\end{align*}
 \begin{align*}y=2x^38x^2+3x12\end{align*}
 \begin{align*}y=2x^33x^25x+10\end{align*}
Graph the functions below and determine the number of real roots. Give at least one factor of each polynomial from the graphed solution.
 \begin{align*}y=x^33x^22x+6\end{align*}
 \begin{align*}y=x^3+x^23x3\end{align*}
 \begin{align*}y=x^3+2x^216x32\end{align*}
 \begin{align*}y=2x^3+13x^2+9x+6\end{align*}
 \begin{align*}y=2x^3+15x^2+4x21\end{align*}
Graph the functions below to determine if \begin{align*}x  1\end{align*} is a factor of the polynomial.
 \begin{align*}y=x^32x^2+3x6\end{align*}
 \begin{align*}y=x^3+3x^22x2\end{align*}
 \begin{align*}y=3x^3+8x^25x6\end{align*}
 \begin{align*}y=x^3+x^210x+8\end{align*}
 \begin{align*}y=2x^3x^23x+2\end{align*}
Indicate the real root(s) on the following quartic graphs:
 \begin{align*}y=x^43x^36x^23\end{align*}
 \begin{align*}y=x^48x^28\end{align*}
 \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*}