1. Problem of the day
Can you get sum of 99 by using all numbers (0-9) and only one mathematical symbols ?

2. TS: Making decisions after reflection and review
Descartes’s Rule of Signs & Bounds
Objective: to be able to use Descartes Rule of Signs and Bounds to graph a polynomial
TS: Making decisions after reflection and review
Warm up:
Using a calculator graph
f(x) = 3x4 + 5x3 – 6x2 + 8x – 3

3. Descartes’s Rule of Signs
Let f(x) = anxn + an-1xn-1 + … +a2x2 + a1x +a0 be a polynomial with real coefficients and a0≠0.
The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer.
The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer

4. Examples: f(x) = 3x4 + 5x3 – 6x2 + 8x – 3 g(x) = 2x3 – 4x2 – 5
Use Descartes’s Rule of Signs to determine the possible numbers of positive and negative real zeros of the function.
f(x) = 3x4 + 5x3 – 6x2 + 8x – 3
g(x) = 2x3 – 4x2 – 5

5. Upper and Lower Bound Rules
Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x – c, using synthetic division.
If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f.
If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f.

6. Sketch the graph.
g(x) = x5 + 3x4 – 8x3 – 24x2 +16x + 48

7. Sketch the graph
h(x) = -x5 – 6x4 – 9x3 + 4x2 + 36x + 24

8. Sketch the graph
m(x) = -2x4 – 2x2 + 24

9. Sketch the graph
f(x) = x4 – 7x2 + 10