Find a degree 3 polynomial with real coefficients having zeros 3 and 2i. and a lead coefficient of 1. Write p in expanded form. Be sure to write the full equation,

Question

Find a degree 3 polynomial with real coefficients having zeros 3 and 2i. and a lead coefficient of 1. Write p in expanded form. Be sure to write the full equation,

Answer 1

x=2i

x^2 -4

x^2 +4 =0

x^2+4 is one factor the other factor is x-3

multipl them together

P(x) = (x-3)(x^2+4) = x^3 -3x^2 +4x -12

That is the most simple P(x),and the only one with lead coefficient = 1

the imaginary zero necessarily comes in conjugate pairs, if 2i is one zero, another zero is-2i

but you can have an infinite number of other functions with the same 3 zeros

P(x) = a(x^3 -3x^2 +4x-12) where a = any real number such as

P(x) = 2x^3 -6x^2 +8x -12 where a=2