# How do you write a polynomial function of least degree that has real coefficients, the following given zeros 3-i,5i and a leading coefficient of 1?

Dec 10, 2016

Any polynomial with real coefficients and imaginary or complex zeros, must have zeros that are conjugate pairs, therefore, the polynomial must have the following factors:

$y = \left(x - 3 + i\right) \left(x - 3 - i\right) \left(x - 5 i\right) \left(x + 5 i\right)$

Multiplying the last two factors is easy; it is the sum of two squares:

$y = \left(x - 3 + i\right) \left(x - 3 - i\right) \left({x}^{2} + 25\right)$

Multiplying the first two factors is a bit more difficult:

$y = \left({x}^{2} - 3 x - i x - 3 x + 9 + 3 i + i x - 3 i - {i}^{2}\right) \left({x}^{2} + 25\right)$

Combine like terms:

$y = \left({x}^{2} - 6 x + 9 - {i}^{2}\right) \left({x}^{2} + 25\right)$

Use the identity ${i}^{2} = - 1$:

$y = \left({x}^{2} - 6 x + 10\right) \left({x}^{2} + 25\right)$

Multiply the remaining factors:

$y = {x}^{4} - 6 {x}^{3} + 10 {x}^{2} + 25 {x}^{2} - 150 x + 250$

Combine like terms:

$y = {x}^{4} - 6 {x}^{3} + 35 {x}^{2} - 150 x + 250$