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Question
Determine the number of positive and negative roots of the equation x9 – 5x8 – 14x7 = 0
Solution
x9 – 5x8 – 14x7 = 0
P(x) = x9 – 5x8 – 14x7.
The number of sign changes is P(x) is 1.
The number of positive roots is 1.
P(– x) = – x9 – 5x8 + 14x7
The number of sign changes is P(– x) is one.
The number of negative zero of P(– x) is 1.
It is clear that 0 is a root of the equation.
∴ The number of the imaginary roots is at least 6.
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