### About Descartes' Rule of Signs:

Descartes' Rule of Signs allows us to determine the possible number of positive real zeros and the possible number of negative real zeros for a polynomial function with real coefficients and a nonzero constant term. This rule will help us to narrow down our choices when looking for zeros of a polynomial function.

Test Objectives
• Demonstrate the ability to find the possible number of real zeros
Descartes' Rule of Signs Practice Test:

#1:

Instructions: State the possible number of positive real zeros and negative real zeros.

a) f(x) = 4x5 + 2x4 + 30x3 + 15x2 - 16x - 8

b) f(x) = 2x5 + 4x4 + 5x3 + 10x2 - 12x - 24

#2:

Instructions: State the possible number of positive real zeros and negative real zeros.

a) f(x) = 4x6 + 16x4 - 25x2 - 100

b) f(x) = 9x6 + 45x4 - 4x2 - 20

#3:

Instructions: State the possible number of positive real zeros and negative real zeros.

a) f(x) = 5x6 - 4x4 - 20x2 + 16

b) f(x) = 6x5 - 9x4 - 34x3 + 51x2 + 20x - 30

#4:

Instructions: State the possible number of positive real zeros and negative real zeros.

a) f(x) = 5x5 + 25x4 + 14x3 + 70x2 - 3x - 15

b) f(x) = 16x6 + 64x4 - 25x2 - 100

#5:

Instructions: State the possible number of positive real zeros and negative real zeros.

a) f(x) = x6 - 64

b) f(x) = 27x7 + 37x4 - 64x

Written Solutions:

#1:

Solutions:

a) Positive Real Zeros: 1, Negative Real Zeros: 4, 2, or 0

b) Positive Real Zeros: 1, Negative Real Zeros: 4, 2, or 0

#2:

Solutions:

a) Positive Real Zeros: 1, Negative Real Zeros: 1

b) Positive Real Zeros: 1, Negative Real Zeros: 1

#3:

Solutions:

a) Positive Real Zeros: 2 or 0, Negative Real Zeros: 2 or 0

b) Positive Real Zeros: 3 or 1, Negative Real Zeros: 2 or 0

#4:

Solutions:

a) Positive Real Zeros: 1, Negative Real Zeros: 4, 2, or 0

b) Positive Real Zeros: 1, Negative Real Zeros: 1

#5:

Solutions:

a) Positive Real Zeros: 1, Negative Real Zeros: 1

b) Positive Real Zeros: 1, Negative Real Zeros: 1
Note: In this case, 0 is a zero but is not positive or negative. This means we would have 3 real zeros.