Quadratic Functions

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Slide 23

a’s are real numbers, an  0, and a0  0.

1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots.

2. f(-x) = -4x3 - 5x2 + 6 has one change-of-signs; thus, f(x) has one negative real root.

Example 1: Descartes’s Rule of Signs

Slide 24

Factor out x; f(x) = x(4x2 - 5x + 6) = xg(x)

1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots.

2. g(-x) = 4x2 + 5x + 6 has zero change-of-signs; thus, g(x) has no negative real root.

Example 2: Descartes’s Rule of Signs

Slide 25

Rational Zero Test

If a’s are integers, every rational zero of f has the form

rational zero = p/q,

in reduced form, and p and q are factors of a0 and an, respectively.

Slide 26

f(x) = 4x3 - 5x2 + 6 p  {1, 2, 3, 6}

q  {1, 2, 4}

p/q  {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4} represents all possible rational roots of f(x) = 4x3 - 5x2 + 6 .

Example 3: Rational Zero Test

Slide 27

Upper and Lower Bound

f(x) is a polynomial with real coefficients and an > 0 with f(x)  (x - c), using synthetic division:

1. If c > 0 and each # in last row is either positive or zero, c is an upper bound.

2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound.

Slide 28

2x3 - 3x2 - 12x + 8 divided by x + 3

-3 2 -3 -12 8 -6 27 -45

2 -9 15 -37

c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots.

Example 4: Upper and Lower Bound