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

1 Answer
Jul 15, 2018

Answer:

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#