Problem of the day

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

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

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

Descartes’s Rule of Signs

Examples:

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

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.

Sketch the graph.

g(x) = x5 + 3x4 – 8x3 – 24x2 +16x + 48

Sketch the graph

h(x) = -x5 – 6x4 – 9x3 + 4x2 + 36x + 24

Sketch the graph

m(x) = -2x4 – 2x2 + 24

Sketch the graph

f(x) = x4 – 7x2 + 10