Polynomial Equations Written In Standard Form
Degree

The highest exponent of a polynomial written in standard form.
Maximum Number of Turns
 Degree minus 1. Remember, this is the MAXIMUM # turns.
Roots









Degree=Number of Roots
Complex and Imaginary Roots occur in conjugate pairs.
Factors of Constant
Possible Rational Roots= 
Factors of Lead Coefficient
Multiplicity of a root = the exponent for the factor
Even multiplicity indicates the graph is tangent to the x-axis. i.e. “bounces off the xaxis”
Odd multiplicity indicates the graph crosses the x-axis. i.e. “it goes through the x-axis”
Roots occur whenever P(x) = 0 or whenever there is a change in the sign of P(x).
Imaginary Roots do not appear on the graph.
Use the calculate zeros function of the calculator to approximate the irrational roots of
the function.
Y-Intercept
 (0, C)
End Behavior
 Odd Degree
Positive Lead Coeff.
Falling to the Left
As x   , f ( x)   
Rising to the Right
As x  , f ( x)  

Even Degree
Positive Lead Coeff.
Rising to the Left
As x   , f ( x)  
Rising to the Right
As x  , f ( x)  
Negative Lead Coeff.
Rising to the Left
As x   , f ( x)  
Falling to the Right
As x  , f ( x)   
Negative Lead Coeff.
Falling to the Left
As x   , f ( x)   
Falling to the Right
As x  , f ( x)   
Greatest Lower Bound (GLB)
 A GLB indicates that no zeros occur to the left of that x value. In other words, we are
not concerned with the function for x values that are smaller than the GLB. A GLB
occurs whenever the signs of the coefficients of the depressed equation alternate. i.e.
The signs in the synthetic division chart alternate whenever read across the table.
3
6 -4 5 -8 2
-7
Least Upper Bound (LUB)
 A LUB indicates that no zeros occur to the right of that x value. In other words, we are
not concerned with the function for x values that are larger than the LUB. A LUB
occurs whenever the signs of the coefficients of the the depressed equation do not
change. i.e. The signs in the synthetic division chart are all positive or all negative
whenever read across the table.
7
6 11 5 4 7
13
Relative Extrema
 Relative Maximum and Minimum points occur at the “peaks and valleys” of the graph.
Since the function continues to infinity, they are not “true” maximum and minimum
points though. Use the calculate maximum and minimum function of the calculator to
determine these points.