Question

Consider the polynomial f(x) = 5x4+ 3x2+ x + 19. How many complex solutions does the polynomial have?

Consider the polynomial f(x)=5x5-8x4+19x2-3. This polynomial has ____real zero's with multiplicity 1 and _____ complex zero's ?

Answer 1

1. there are 4 complex solutions
2. 3 real zeros and 2 complex zeros

Explanation:

1. Descarte's rule of signs tells you there are 0 positive real roots and 0 or 2 negative real roots. (for positive x, signs are ++++ so have no changes; for negative x, signs are ++-+, so have 2 changes.) A graph shows no real roots.

2. There are 3 sign changes in the given polynomial, so 3 or 1 positive real roots. When the sign of x is changed, there are 2 sign changes, so 0 or 2 negative real roots. A graph shows 2 negative and one positive real root (for a total of 3), so the remaining 2 roots are complex.