How do you find a third degree polynomial given roots 3 and 2-i?

1 Answer
Hammer
Jun 18, 2018

We are given roots

x_1=3
x_2=2-i

The complex conjugate root theorem states that, if P is a polynomial in one variable and z=a+bi is a root of the polynomial, then bar z=a-bi, the conjugate of z, is also a root of P.

As such, the roots are

x_1=3
x_2=2-i
x_3=2-(-i)=2+i

From Vieta's formulas, we know that the polynomial P can be written as:

P_a(x)=a(x-x_1)(x-x_2)(x-x_3)

Where a is a constant.

:. P_a(x) = a(x-3)(x-2+i)(x-2-i)

:. P_a(x) =a(x^3-7x^2+17x-15)

One such polynomial would be

P_1(x) = x^3-7x^2+17x-15

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