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

1 Answer
Jul 15, 2018

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