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

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Dr. Claude S. Moore Danville Community College

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Presentation transcript:

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

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

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

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

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