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
#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
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#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
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#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
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#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
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#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
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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
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#2:
Solutions:
a) Positive Real Zeros: 1, Negative Real Zeros: 1
b) Positive Real Zeros: 1, Negative Real Zeros: 1
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#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
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#4:
Solutions:
a) Positive Real Zeros: 1, Negative Real Zeros: 4, 2, or 0
b) Positive Real Zeros: 1, Negative Real Zeros: 1
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#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.
