1. Polynomial
Functions
2.3

2. A polynomial function is a function of the form:
Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 1, 2 …; include 0
A polynomial function is a function of the form:
n must be a positive integer or 0
All of these coefficients are real numbers
The degree of the polynomial is the largest power on any x term in the polynomial.

3. Determine which of the following are polynomial functions
Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.
A polynomial of degree 4.
We can write in an x0 since this = 1.
x 0
A polynomial of degree 0.
Not a polynomial because of the square root since the power is NOT an integer
Not a polynomial because of the x in the denominator since the power is negative

4. Graphs of polynomials are smooth and continuous.
No gaps or holes, can be drawn without lifting pencil from paper
No sharp corners or cusps
This IS the graph of a polynomial
This IS NOT the graph of a polynomial

5. Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the leading coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behavior determined by the highest powered term.
left hand behavior: rises to the left
right hand behavior: rises to the right

6. Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the leading coefficient is negative.
turning points in the middle
left hand behavior: falls to the left
right hand behavior: falls to the right

7. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the leading coefficient is positive.
turning points in the middle
right hand behavior: rises to the right
left hand behavior: falls to the left

8. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the leading coefficient is negative.
turning points in the middle
left hand behavior: rises to the left
right hand behavior: falls to the right

9. CONCLUSION and LEFT RIGHT HAND BEHAVIOR OF A GRAPH
The degree of the polynomial along with the sign of the coefficient of the term with the highest power tells us about the left and right hand behavior of a graph.

10. Example: – 2 is a zero of multiplicity 2
Can you find the zeros of the polynomial?
There are repeated factors.
(x – 1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and solve we get x = 1. We then say that x = 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times).
What are the other zeros and their multiplicities?
– 2 is a zero of multiplicity 2
3 is a zero of multiplicity 1

11. Conclusion and Application of Multiplicity of Zeros
If we know the multiplicity of the zero, it tells us whether the graph
crosses the x axis at this point
(odd multiplicities CROSS), or
whether it just touches the axis and turns
and heads back the other way
(even multiplicities TOUCH).

12. Steps for Graphing a Polynomial
Determine left and right hand behavior by looking at the highest power on x and the sign of that term.
***Determine maximum number of turning points in graph by subtracting 1 from the degree.***
Find and plot y intercept by putting 0 in for x
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Determine multiplicity of zeros.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide. Use additional points for the final graph estimation.

13. Find and plot y intercept by putting 0 in for x
Let’s graph:
Determine left and right hand behavior by looking at the highest power on x and the sign of that term.
Find and plot y intercept by putting 0 in for x
Determine maximum number of turns in graph by subtracting 1 from the degree.
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.
Determine multiplicity of zeros.
0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses)
Zeros are: 0, 3, -4
Multiplying out, highest power would be x4
Degree is 4 so maximum number of turns is 3
Here is the actual graph. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value.

14. Practice Exercise:
Graph by hand

15. What is we thought backwards
What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros –1, 2 and 3.
What would the function look like in factored form to have the zeros given above?
Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one.