# Math Help - Imaginary roots

1. ## Imaginary roots

Let P(z) = z^3 + az^2 + bz + c, where a, b and c E R. Two of the roots of P(z) = 0 are -2 and (-3+2i). Find the value of a, b and c

2. Originally Posted by Adam We
Let P(z) = z^3 + az^2 + bz + c, where a, b and c E R. Two of the roots of P(z) = 0 are -2 and (-3+2i). Find the value of a, b and c

You might want to let z = x + iy so it's easier to distinguish between the real and complex parts.

To find the third root, remember that if a complex number is a root, then its conjugate is also a root.

Then P(Z) will equal the product of its factors, and (remembering to match real with real, imaginary with imaginary!) solve for a,b,c.

3. Originally Posted by Unenlightened
You might want to let z = x + iy so it's easier to distinguish between the real and complex parts.

To find the third root, remember that if a complex number is a root, then its conjugate is also a root. Mr F adds: Provided the coefficients of the polynomial are all real (which they are in this case).

Then P(Z) will equal the product of its factors, and (remembering to match real with real, imaginary with imaginary!) solve for a,b,c.
And note that a fast way to expand $(z - (\alpha + i \beta))(z - (\alpha - i \beta))$ is to note that it's equal to $(z - \alpha - i \beta)(z - \alpha + i \beta) = ([z - \alpha] - i \beta)([z - \alpha] + i \beta) = (z - \alpha)^2 + b^2 = \, ....$