State the possible number imaginary zeros for the function.

Question

x^5+4x^4−3x^3+7x^2−2x+10

Answer 1

The fundamental theorem of algebra states there are the same number of total zeros as the degree of the polynomial so there will be 5 total zeros.

Descartes Rule of Signs says 1) there will be the same number of positive Real zeros as there are sign changes, or increments of 2 less. There are 4 sign changes:

So there will either be 4 positive Real zeros, or there will be 2 positive Real zeros or there will be no positive Real zeros.

Descartes Rule of Signs also says 2) there will be the same number of negative Real zeros as there are sign changes for f(-x), or increments of 2 less. There 1 sign change:

So there will be one negative Real zero.

So in summary we have:

So there can be 4 possible imaginary (complex) zeros, or 2 possible imaginary (complex) zeros, or 0 possible imaginary (complex) zeros.