How do you write a polynomial function with minimum degree whose zeroes are 5 and 3+2i?

1 Answer
Nov 11, 2017

see below

Explanation:

a polynomial" "P(x) P(x) with minimum degree.

We have the coefficients real, therefore all complex roots occur in conjugate pairs

we have zeros

x=5=>(x-5)x=5(x5)" "is a factor of P(x)P(x)

x=3+3i=>(x-(3+2i))x=3+3i(x(3+2i))a factor

:.x=3-2i=>(x-(3-2i))a factor

P(x)=(x-5)(x-(3+2i))(x-(3-2i))

P(x)=(x-5)(x^2-x(3-2i)-x(3+2i)+(3+2i)(3-2i))

=(x-5)(x^2-3x+cancel(2x i)-3xcancel(-2x i)+9+4)

=(x-5)(x^2-6x+13)

=x^3-6x^2+13-5x^2+30x-65

x^3-11x^2+30x-52