Question

What are the possible numbers of positive real, negative real, and complex zeros of f(x) = −7x4 − 12x3 + 9x2 − 17x + 3?

Answer 1

Now, recall Descartes Rule of Signs.  Check the picture below.

the +x part, gives us the positive Real zeros, and it depends on how many times the sign changes from term to term, notice in the picture, it changed 3 times, so the positive real ones are 3, or 3-2, namely 1, so 3 or 1.

the negative real ones, come from using -x as the argument on f(x), and as you can see in the picture, there was only 1 sign change, meaning the negative real zeros are only 1.

since, based on the fundamental theorem of algebra the polynomial has 4 roots at most then,

3 positive real ones, and 1 negative real one, no complex ones

or

1 positive real one, and 1 negative real one, and 2 complex ones

bear in mind that complex always come in pairs, never alone.