How do you use the rational roots theorem to find all possible zeros of #x^4-5x^3-5x^2+23x+10#?

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Answer 1 · 2016-03-19T12:08:08

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Answer 2 · 2016-03-19T13:53:35

#-2,5, 1+-sqrt(2)#

By the rational root theorem the rational roots of the polynom can be:

#+-1, +-2 or +-5#

If you try the roots, you will confirm that the real roots are -2 and 5:

Now you have to divide the polynom by :

#(x+2)(x-5)=x^2-3x-10#

#(x^4-5x^3-5x^2+23x+10)/(x^2-3x-10)=#

#x^2-2x-1#

Use the quadratic formula to solve this part:

#x=(2+-sqrt(4+4))/2=(2+-sqrt(8))/2=(2+-2sqrt(2))/2#

#x=1+-sqrt(2)#