conjugates and real zeros

if z=a+bi, then the conjugate of z is written z^, where z^ =a-bi.
For example, if z=-5+2i, then z^=-5-2i.

finding all zeros given one zero

The theorem on conjugate zeros helps predict the number of real zeros of polynomial functions with real coefficients. a polynomial function with real coefficients of odd degree n, where n greater or equal to 1 must have at least one real zero, since zeros of the form a+bi where b not equal to zero occur in conjugate pairs. on the other hand a polynomial function with real coefficients of even degree n may have no real zeros.

find all zeros of f(x)=x^4 -7x^3+18x^2-22x+12 given that 1-i is a zero of f(x)

solution: since the polynomial function has only real coefficients and since 1-i is a zero , by the conjugate zeros theorem 1+i is also a zero. to find the remaining zeros, first use synthetic division to divide the original polynomial by x-(1-i).
 _______________________________
1-i) 1       -7        18           -22        12
                1-i       -7+5i      16-6i    -12
      1       -6-i      11+5i      -6-6i       0

by the factor theorem, since x=1-i is a zero of f(x), x-(1-i) is a factor, and f(x) can be written as
 f(x)=[x-(1-i)][X^3+(-6-i)x^2+(11+5i)x+(-6-6i)].

we know that x=1+i is also a zero of f(x), so f(x)=[x-(1-i)][x-(1+i)]q(x).