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 ...