How do you form a polynomial function whose zeros are 3 and 4 + i?

1 Answer
Feb 14, 2016

Answer:

Convert zeros to factors and multiply out to find the simplest possible polynomials.

Explanation:

If #a# is a zero of a polynomial #f(x)#, then #(x-a)# is a factor and vice versa.

So the simplest polynomial with these zeros is:

#(x-3)(x-(4+i)) = x^2-(7+i)x+(12+3i)#

If you want Real coefficients, then the Complex conjugate #4-i# must also be a zero and we find the simplest polynomial is:

#(x-3)(x-(4+i))(x-(4-i))#

#=(x-3)((x-4)^2-i^2)#

#=(x-3)(x^2-8x+17)#

#=x^3-11x^2+41x-51#

Any polynomial with these zeros must be a multiple (scalar or polynomial) of these 'simplest' polynomials.