View Single Post

Mentor

Quote by bob1182006 					 				 				1. The problem statement, all variables and given/known data Find the roots of: $$x^5-1=0$$  2. Relevant equations Polynomial long division.  3. The attempt at a solution $$x^5-1 = (x-1)(x^4+x^3+x^2+x+1) = 0$$ $$x^4+x^3+x^2+x+1 = (x^2+1)^2+x^3+x-x^2$$ $$(x^2+1)^2+x^3+x-x^2 = (x^2+1)^2+x(x^2+1)-x^2$$  Stuck at this point, I just can't seem to factor out something useful.  I know all of the roots are complex but I need to be able to solve the problem without a computer.
Four of the roots of x5 - 1 = 0 are complex and one is real (x = 1). The complex roots are located around the unit circle at 72 deg, 144 deg, 216 deg, and 288 deg. These can be represented in rectangular form, with the first one being cos(72 deg) + i sin(72 deg). The others can be represented similarly. I don't know if there's going to be a way to factor your fourth-degree factor.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9426074028015137, "perplexity_flag": "head"}