Given that 3−5i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4−7x3+10x2+146x−1020

Since 3- 4i  is a zero, by the conjugate root theorem, so must 3 + 4 i be a zero.  

This implies that f(x) has the form

f(x) = [x -(3 -4i) ][x - (3 + 4i)]  Q(x)   where Q is a second order polynomial

multiplying things out:   f(x) = (x2 -6x +25) Q(x).  

This means that Q(x) =  (x4 - 2 x3 - 4 x2 +130 x -125)/(x2 -6 x +25)

Either long division or synthetic division can be used to work out Q

The result is   Q(x) = x2 + 4x -5  = (x + 5)_ (x -1)

The complete factorization (in terms of real coefficients) is

f(x) = (x2 - 6 x +25 )( x +5) (x -1)

Step-by-step explanation:

homozygous. one is homozygous, while two is heterozygous.