Question

Given that the polynomial function (below) has the given zero, find the other zeros? `f(x) = x^4 - 5x^3 + 7x^2 - 5x + 6; -i`

I know the answers are `i, 2, 3` but I don't know how to find them.

Answer 1

The general factored form of a quartic equation is:

`y = k(x-r_1)(x-r_2)(x-r_3)(x-r_4)" [1]"`

Where `r_1,r_2,r_3, and r_4` are the roots of the equation.

Explanation:

Given: `f(x) = x^4 - 5x^3 + 7x^2 - 5x + 6; -i`

We can substitute `-i` for `r_1` into equation [1]:

`y = k(x+i)(x-r_2)(x-r_3)(x-r_4)" [1.1]"`

Because all of the coefficients are real, we know that imaginary roots must exist in conjugate pairs, therefore, `i` must, also, be a root. Substitute `i` for `r_2` in equation [1.1]:

`y = k(x+i)(x-i)(x-r_3)(x-r_4)" [1.2]"`

Because the leading coefficient of the given equation is 1, we know that `k = 1`, therefore, remove k from equation [1.2]:

`y = (x+i)(x-i)(x-r_3)(x-r_4)" [1.3]"`

We know that `(x-i)(x+i) = x^2+1`, therefore, the original polynomial must be evenly divisible by `x^2+1`:

`(x^4 - 5x^3 + 7x^2 - 5x + 6)/(x^2+1) = x^2-5x+6 larr` the factors of this quadratic must contain the remaining roots:

`x^2-5x+6 = (x-2)(x-3)`

`y = (x+i)(x-i)(x-2)(x-3)" [1.4]"`

The roots are: `-i, i, 2, and 3`