How do you find all rational zeroes of the function using synthetic division #f(x)=x^4+x^3+x^2-9x-10#?

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Answer 1 · 2018-01-14T16:34:53

#x_1=-1-2i#, #x_2=-1+2i#, #x_3=-1# and #x_4=2#

After using Rational Roots Test, #x=-1# is a root of the polynomial. Hence #x+1# is multiplier of it. Consequently,

#x^4+x^3+x^2-9x-10#

=#x^3*(x+1)+(x+1)*(x-10)#

=#(x+1)*(x^3+x-10)#

=#(x+1)*(x^3-8+x-2)#

=#(x+1)[(x-2)*(x^2+2x+4)+x-2]#

=#(x+1)(x-2)(x^2+2x+5)#

Hence roots of #f(x)# are #-1, 2, -1+2i# and #-1-2i#