Question

How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 4, 4, 2+i?

Answer 1

`f(x) = x^4-12x^3+53x^2-104x+80`

Explanation:

Assuming you want a polynomial with real coefficients, the complex conjugate `2-i` must also be a zero and the monic polynomial of least degree is:

`f(x) = (x-4)(x-4)(x-(2+i))(x-(2-i))`

`color(white)(f(x)) = (x^2-8x+16)((x-2)+i)((x-2)-i)`

`color(white)(f(x)) = (x^2-8x+16)((x-2)^2-i^2)`

`color(white)(f(x)) = (x^2-8x+16)(x^2-4x+4+1)`

`color(white)(f(x)) = (x^2-8x+16)(x^2-4x+5)`

`color(white)(f(x)) = x^4-12x^3+53x^2-104x+80`

Any polynomial in `x` with these zeros will be a multiple (scalar or polynomial) of this `f(x)`.