Question

How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are -2, -3, i, -i?

Answer 1

`P(x) = (x+2)(x+3)(x^2+1)`
`" " = x^4+5x^3+7x^2+5x+6`

Explanation:

Suppose the polynomial is `P(x)`

By the factor theorem, if `x=a` is root of `P(x)=0`, then `x=a` is a factor of `P(x)`

We have the following roots of `P(x)=0`

`x=-2,-3,i,-i`

Hence, the following are factors of `P(x)`

`(x+2), (x+3), (x-i), (x+i)`

Hencde, we can write the polynoimal of least degree as the product of these factors (any higher degree polynomial would have additional roots)

`P(x) = A(x+2)(x+3)(x-i)(x+i)`

We want our polynomial to have leading coefficient `1=>A=1`, and also to have real coefficients, so let us multiply out the complex factors (as they are conjugates we will get a real product)

`(x-i)(x+i) = x^2 + ix - ix -i^2 = x^2+1`

Thus we have:

`P(x) = (x+2)(x+3)(x^2+1)`
`" " = x^4+5x^3+7x^2+5x+6`