Question #33aa8

1 Answer
Jim G.
Jan 5, 2018

#(cosx+1/2+1/2sqrt3i)(cosx+1/2-1/2sqrt3i)#

Explanation:

#"let "cosx=x#

#"then "#

#cos^2x+cosx+1-=x^2+x+1#

#"find the roots of "x^2+x+1#

#"using the "color(blue)"quadratic formula"#

#"with "a=1,b=1" and "c=1#

#•color(white)(x)x=(-b+-sqrt(b^2-4ac))/(2a)#

#color(white)(xxx)=(-1+-sqrt(1-4))/2=(-1+-sqrt-3)/2#

#rArrx=-1/2+-1/2sqrt3i#

#"factors are"#

#(x-(-1/2-1/2sqrt3i))(x-(-1/2+1/2sqrt3i))#

#=(x+1/2+1/2sqrt3i)(x+1/2-1/2sqrt3i)#

#"convert x back to "cosx#

#rArrcos^2x+cosx+1#

#=(cosx+1/2+1/2sqrt3i)(cosx+1/2-1/2sqrt3i)#

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