Question

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?

Answer 1

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 = (x - 3 + i)(x - 3 - i)(x - 5i)(x + 5i)`

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

`y = (x - 3 + i)(x - 3 - i)(x^2 + 25)`

Multiplying the first two factors is a bit more difficult:

`y = (x^2 - 3x - ix -3x + 9 + 3i + ix -3i - i^2)(x^2 + 25)`

Combine like terms:

`y = (x^2 - 6x + 9 - i^2)(x^2 + 25)`

Use the identity `i^2 = -1`:

`y = (x^2 - 6x +10)(x^2 + 25)`

Multiply the remaining factors:

`y = x^4 - 6x^3 +10x^2 + 25x^2 -150x + 250`

Combine like terms:

`y = x^4 - 6x^3 + 35x^2 -150x + 250`