Question

How many complex zeros does the polynomial function have? F(x)=2x^4 +5x^3 - x^2 +6x-1

Answer 1

Answer:

Two complex roots.

Step-by-step explanation:

F(x)=2x^4 +5x^3 - x^2 +6x-1

is a polynomial in x of degree 4.

Hence F(x) has 4 roots.  There can be 0 or 2 or 4 complex roots to this polynomial since complex roots occur in conjugate pairs.

Use remainder theorem to find the roots of the polynomial.

F(0) = -1 and F(1) = 2+5-1+6-1 = 11>0

There is a change of sign in F from 0 to 1

Thus there is a real root between 0 and 1.

Similarly by trial and error let us find other real root.

F(-3) = -1 and F(-4) = 94

SInce there is a change of sign, from -4 to -3 there exists a real root between -3 and -4.

Other two roots are complex roots since no other place F changes its sign