find the rational roots of the polynomial 2x^3+3x^2-11x-6 .

Answers (1)

solution: Let f(x)=2x^3+3x^2-11x-6

clearly, f(x) is  cubic polynomial  with integer coefficients. if    fracbc    is a rational roots in lowest term , thenthe value of   b   are limited to the factors of 6 which are pm1, pm2,pm3,pm6 ;

and the value of c are limited to the factors of  which are pm1,pm2.  Hence, the  possible  rational roots of f(x) are 

                      pm1,pm2,pm3,pm6,pmfrac12,pmfrac32

we observe that 

                      f(2)=2	imes 8+3	imes4-11	imes2-6=0

                      f(-3)=2	imes-27+3	imes9-11	imes-3-6=-54+27+33-6=0

  and              f(-frac12)=2	imes-frac18+3	imesfrac14-11	imes-frac12-6=-frac14+frac34+frac112-6=0

Hence ,  2,-3  and -frac12  are rational roots of f(x).

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