1. Polynomial Curve
Sketching

2. Warm up
When the polynomial 2x2 + bx - 5 is divided by x - 3, the remainder is 7.
(a) Determine the value of b.

3. Warm up
Solve by factoring the polynomial completely.
(a) x 3 + 5x 2 + 2x - 8 = 0

4. Use your TI 83 to find the roots of the
polynomial.
x3 - 2x2 - 5x + 6
Answer x = -2, 1, 3

5. Degree n of a polynomial is odd
The function has opposite behaviour
If the leading coefficient is >0
The graph rises to the right and falls to
the left
3 2
x +2x +1
If the leading coefficient is <0
The graph rises to the left and falls to
the right
3 2
-2x +2x +1

6. When the degree n of a polynomial is even,
then the graph has similar behaviour on the
left as on the right
If the leading coefficient >0 the graph rises
on the left and rises on the right
4 3 2
x + x - 2x + x + 1
If the leading coefficient <0 the graph falls on
the left and falls on the right
-x4 - 2x3 + 2x2 + x + 1

7. Graphing Polynomial Functions
Appearance ƒ(x) = x n
Where n is even, the Where n is odd, the
graph looks like this: graph looks like this:

8. Graphing Polynomial Functions
Roots
The maximum number of roots for any polynomial function is
equal to the degree of the function.
Examples: Cubic Quartic Quintic
ƒ(x) = x 3 ƒ(x) = x 4 ƒ(x) = x 5
max. # of roots 3 4 5

9. Graphing Polynomial Functions
Sketching
Step 1: Find the y-intercept (let x = 0)
Step 2: Find all roots. (Use rational roots theorem if necessary.)
Step 3: Determine the sign of the function over the intervals
defined by the roots.
Step 4: Sketch the graph

10. Factor the polynomial completely. Sketch the graph.
ƒ(x) = x 3+ 5x 2 + 2x - 8

12. Sketch the graph of