Question

Use Descartes' Rule of Sign to determine the possible types of zeros of the function:
f(x) = 2x3 - 24x2 + 94x - 120

Answer 1

The Rule of Signs is pretty cool! Here's how you use it.

Write your polynomial in decreasing order of exponents:

\[f(x) = 2x^3-24x^2+94x-120\]
Now scan along along from left to right, and count how many times the sign changes. For your polynomial, it goes + - + - so there are 3 sign changes. This gives us the maximum possible number of positive zeros. It doesn't determine the number of positive zeros exactly, because we may have pairs of complex zeros, but the count of sign changes gives us the maximum possible number of positive zeros, possibly adjusted down by a multiple of 2.

Now we rewrite the function as f(-x):
\[f(-x) = 2(-x)^3 - 24(-x)^2+94(-x) -120 = -2x^3-24x^2-94x-120\]Scan from left to right again, counting the sign changes. This will give us the maximum number of negative zeros. Again, it may be decreased downward by a multiple of 2 to account for complex zeros.

Finally, keep in mind that you must always have a count of zeros equal to the highest order exponent in your polynomial. For your polynomial, the highest order exponent is 3, so there are 3 zeros. Some of them may be the same value (multiplicity).

Answer 2

So if you follow those directions, what do you conclude about the zeros of that polynomial?