1. 5.3
Polynomial Functions

2. Polynomial functions-
A polynomial in one variable is a function in the form
f(x) = 5x8 + 3x7 – 8x6 + … + 2x1 + 11
an is the leading coefficient
n is the degree of the polynomial
a0 is the constant term

3. Type of polynomial DEGREE 1 2 3 4 TYPE Constant Linear Quadratic Cubic1 2 3 4
TYPE
Constant
Linear
Quadratic
Cubic
Quartic

4. Example 1 : Answer these questions for the following functions.
Is it a polynomial in one variable? Explain.
What is the degree of the polynomial
What is the leading coefficient of the polynomial?
What is the constant (y-intercept) of the polynomial?
What type of polynomial is it?
a. f(x) = 3x2 + 3x3 – 7x4 + x – 6
b. f(x, y) = 4x5 – 14xy4 + x – 6
c. f(y) = (5 – 4y)(6 + 2y)

5. To evaluate a function, you need to plug a value into the function.
Example 2: a) f(x) = 3x5 – x4 – 5x + 10 Find f(–2)

6. To evaluate a function, you need to plug a value into the function.
Example 2: b) f(x) = x5 – x4 – 5x + 10 and g(x) = 2x3 – x – 5 Find f(3d)

7. To evaluate a function, you need to plug a value into the function.
Example 2: c) f(x) = x5 – x4 – 5x + 10 and g(x) = 2x3 – x – 5 Find g(p2)

8. End Behavior: Even Positive Negative Odd Degree Leading Coefficient
Left-hand Behavior
Right-hand Behavior
Example Picture
Even
Positive
Negative
Odd

9. Example 5: Describe the end behavior for each polynomial.
a. f(x) = 3x5 – x4 – 5x + 10
b. f(x) = 3x3 – 4x6 + 2x

10. Real Zeros
The x-intercepts of the function.

11. Degree of polynomial
Turning points – the points on a graph where the function changes its vertical direction (up/down).
If the degree of the polynomial is n, there will be at most n – 1 turning points
Ex) If a polynomial has 8 turning points then the degree is ________.

12. Example 6 Do the following for the graph: a) Describe the end behavior
b) Determine if it’s an odd-degree or an even-degree function
c) State the number of real zeros.
Answer:
as x → –∞, f(x) → –∞ and as x → +∞, f(x) → –∞
It is an even-degree polynomial function.
The graph does not intersect the x-axis, so the function has no real zeros.

13. Example 7 Do the following for the graph: a) Describe the end behavior
b) Determine if it’s an odd-degree or an even-degree function
c) State the number of real zeros.
Answer:
As x → –∞, f(x) → –∞ and as x → +∞ , f(x) → +∞
It is an odd-degree polynomial function.
The graph intersects the x-axis at one point, so the function has one real zero.