1. Topic 8-3 Polynomials, Linear Factors & Zeros
Exploring the Significant Points
of Polynomial Graphs
(graphs with all rational roots)

2. The significant points on a polynomial graph are the x-intercepts (Real Roots), the y-intercept, all relative minima and all relative maxima (turning points).
f(x) = 3x4 + 9x3 – 12x2
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
First, what do we know about the graph’s end behaviors?
The lead coefficient is positive and it is a polynomial of degree 4 (even)…
It’s right side end behavior will be rising since the lead is positive (y approaches positive infinity as the x approaches positive infinity). It’s left side end behavior will also be rising since it is an even degreed function and they do the same thing on both sides (so y approaches positive infinity as x approaches negative infinity).

3. f(x) = 3x4 + 9x3 – 12x2
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
It’s right side end behavior will be rising (y approaches positive infinity as the x approaches positive infinity).
It’s left side end behavior will also be rising since it is an even degreed function (so y approaches positive infinity as x approaches negative infinity).
What do we know about the graph’s ZEROS?
It has at most 4 zeros (or x-intercepts)… degree of 4!
So type it into [y=] and [GRAPH].
Picture what the graph is doing OFF Screen…
Based on what we know:

4. f(x) = 3x4 + 9x3 – 12x2 2 Relative Minima
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
It’s right side end behavior will be rising (y approaches positive infinity as the x approaches positive infinity).
It’s left side end behavior will also be rising since it is an even degreed function (so y approaches positive infinity as x approaches negative infinity).
It appears to have 2 zeros where the graph crosses the x-axis and a zero where the graph hits it and runs away (like a quadratic with a double root). So it has a zero with multiplicity 2 and two singular zeros.
How many local minimums and maximums does it have?
Based on the behavior, it has two minima and one maximum…
It has at most 4 zeros (or x-intercepts).
1 Relative Maximum

5. f(x) = 3x4 + 9x3 – 12x2
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
It’s right side end behavior will be rising (y approaches positive infinity as the x approaches positive infinity).
It’s left side end behavior will also be rising since it is an even degreed function (so y approaches positive infinity as x approaches negative infinity).
The significant points from left to right are the first root, the first minimum, the maximum (also the second root and the y-intercept), the second minimum and the third root.
Use your Calculator to find those five points…
It has at most 3 zeros (or x-intercepts), one of which is multiplicity two. It has 1 local MAX and 2 local MIN’s.

6. f(x) = 3x4 + 9x3 – 12x2 f(x) = 3x2(x + 4)(x – 1)
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
The significant points from left to right are the first root, the first minimum, the maximum (also the second root and the y-intercept), the second minimum and the third root.
Find the roots…
[2nd] [calc] [zero]
set your left & right boundaries and hit [enter] when you get to “Guess?”
It is at (–4, 0)
So in Root Form the polynomial has a factor of (x + 4)
The other zeros are at (0, 0) & (1, 0). The (0, 0) root has multiplicity 2! So factors of x2 and (x – 1)… Btw, a = 3
So the Root Form of the polynomial is
f(x) = 3x2(x + 4)(x – 1)

7. f(x) = 3x4 + 9x3 – 12x2 f(x) = 3x2(x + 4)(x – 1)
List the coordinates of the significant points for the function. Use the x-intercepts to write the function in Root Form. Draw a rough sketch of each function and label the significant points on the graph.
The significant points from left to right are the first root, the first minimum, the maximum (also the second root and the y-intercept), the second minimum and the third root.
We already know the local maximum is (0, 0) since it is also an x-intercept…
Now let’s find the two local minima.
We will have to use our calculator twice! [2nd] [calc] [3: minimum]
Be careful setting the left & right boundaries when you “surround” your point.
First minimum is at (-2.93, ). The second is (0.68, -2.08) rounded to two decimal places.
Root Form:
f(x) = 3x2(x + 4)(x – 1)
X-intercepts:
(-4, 0), (0, 0) & (1, 0)

8. Sketch the graph onto your paper and label the points!
Root Form:
f(x) = 3x2(x + 4)(x – 1)
X-intercepts:
(-4, 0), (0, 0) & (1, 0)
Y-intercept:
(0, 0)
Local Maxima:
Local Minima:
(-2.93, ) & (0.68, -2.08)
(0, 0)
(-4, 0)
(1, 0)
(0.68, -2.08)
(-2.93, )

9. Practice:
WS - Graphs of Polynomial Functions – Finding the Significant Points
L08-3 WS Graphs of Polynomial Functions.pdf