Question

What are the possible number of positive, negative, and complex zeros of f(x) = –3x4 + 5x3 – x2 + 8x + 4 ?

Answer 1

It equals the highest degree of your polynomial.

Answer 2

descartes' ?

Answer 3

All zeros are complex, meaning they are of the form a+bi. Sometimes b=0, which eliminates the imaginary component, making it appear to be only real. So the number of complex roots of a polynomial is equal to the highest degree. I missed that you asked about positive and negative zeros. That would use Descartes' Rule of Signs as @shubhamsrg suggested.

Answer 4

Descarte's Rule says to count how many times the signs change from term to term in your polynomial. Then the number of positive zeros is either equal to that number or equal to that number minus even integers. I count 3 sign changes, so that would mean that there are either 3 or 3-2=1 positive real zeros.

Answer 5

The next step is to put a -x into your polynomial everywhere there is an x. After you see how that negative would work out through the various powers, count the sign changes in this new f(-x). The number of negative real zeros is equal that number or that number minus even integers. I counted 1. That would mean that there is 1 negative zero.

Answer 6

Since they asked for the possible numbers... I would say there are 3 positive real zeros possible, 1 negative real zero possible. And the number of complex roots is equal to the degree... do you know what that is?