Question

Find the number of possible positive real zeros of f(x)=8x^4-72x^3+144x^2 a. four c. two or none b. three d. none Please select the best answer from the choices provided A B C D

Answer 1

c. two or none

Explanation:

The signs of the coefficients are ...

+ - +

so there are two sign changes: + to -, and - to +. Descartes' rule of signs tells you to interpret this to mean there are 2 or 0 possible positive real roots.

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Pairs of roots are possibly complex, so the number of sign changes can be reduced by any even number to take that possibility into account. That is why 2 sign changes can mean either 2 or 0 positive real roots.

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A graph shows there to be 2 positive real roots and a root at x=0 with multiplicity 2. The latter should be no surprise, since x^2 is a factor of f(x).