Question

How do I use a graphing calculator to find the complex zeros of `x^4-1?`

Answer 1

You can't easily find complex zeros from a graphing calculator. See answer below.

Complex zeros: `" "x = +- i`

Explanation:

Given: `x^4 - 1`

Graph of the function: `x^4 - 1`:

graph{x^4 - 1 [-5, 5, -5, 5]}

You can't find complex zeros from a graphing calculator, but you can find the real zeros and then use synthetic division or long division to find the complex zeros:

The graph shows that there are zeros at `x = -1, x = 1`

Using synthetic division put the values in the following order:

`x ="coefficient of "x^4, x^3, x^2, x, "constant"`
`ul(-1)| 1" "0" "0" "0" "-1`
`ul(+" "-1" "1" " -1 " "1" ")`
`" "1" "-1" "1" "-1" "0`

These values represent `"coefficients of "x^3, x^2, x, "constant"` and the remainder `= 0`

`x^4 - 1 = (x+1)(x^3-x^2+x-1)`

2nd division of `x^3-x^2+x-1` with `x = 1`:

`x ="coefficient of "x^3, x^2, x, "constant"`
`ul(1)| 1" "-1" "1" "-1`
`ul(+" "1" "0" " 1 ")`
`" "1" "0" "1" "0"`

These values represent `"coefficients of "x^2, x, "constant"` and the remainder `= 0`

`x^4 - 1 = (x+1)(x-1)(x^2+1)`

The complex zeros come from `x^2 + 1`

`x^2 =-1`

`x = +- sqrt(-1) = +- i`

`x^4 - 1 = (x+1)(x-1)(x + i)(x - i)`

Complex zeros: `" "x = +- i`