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

Jul 15, 2018

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

Complex zeros: $\text{ } x = \pm i$

#### 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:

$x = \text{coefficient of "x^4, x^3, x^2, x, "constant}$
$\underline{- 1} | 1 \text{ "0" "0" "0" } - 1$
$\underline{+ \text{ "-1" "1" " -1 " "1" }}$
$\text{ "1" "-1" "1" "-1" } 0$

These values represent $\text{coefficients of "x^3, x^2, x, "constant}$ and the remainder $= 0$

${x}^{4} - 1 = \left(x + 1\right) \left({x}^{3} - {x}^{2} + x - 1\right)$

2nd division of ${x}^{3} - {x}^{2} + x - 1$ with $x = 1$:

$x = \text{coefficient of "x^3, x^2, x, "constant}$
$\underline{1} | 1 \text{ "-1" "1" } - 1$
$\underline{+ \text{ "1" "0" " 1 }}$
$\text{ "1" "0" "1" "0}$

These values represent $\text{coefficients of "x^2, x, "constant}$ and the remainder $= 0$

${x}^{4} - 1 = \left(x + 1\right) \left(x - 1\right) \left({x}^{2} + 1\right)$

The complex zeros come from ${x}^{2} + 1$

${x}^{2} = - 1$

$x = \pm \sqrt{- 1} = \pm i$

${x}^{4} - 1 = \left(x + 1\right) \left(x - 1\right) \left(x + i\right) \left(x - i\right)$

Complex zeros: $\text{ } x = \pm i$