Question

Write a polynomial functionſ of least degree that has rational coefficients, a leading coefficient of 1, and the zeros 1,3, and

2-i.

Answer 1

f(x) =
`x^(4) - 8x^(3) + 24x^(2) - 32x + 15`

Explanation:

First of all, if 2 - i is a zero, then so is 2 + i

All you need to do now is multiply (x - 1)(x - 3)(x - 2 + i)(x - 2 - i)

I will do part of that multiplication

(x - 2 + i)(x - 2 - i) =
`x^(2)` - 2x - xi - 2x + 4 + 2i + xi - 2i -
`i^(2)`

=
`x^(2)` - 4x + 5

So, now (x - 1)(x - 3)(
`x^(2)` - 4x + 5) =
`x^(4) - 8x^(3) + 24x^(2) - 32x + 15`

I will let you finish that multiplication. I did some and you can do some. OK?