If f (x)=2 x^3+ m x^2-13 x+ n and 2,3 are the roots of the equation f (x)=0 then the value of 4 ~m +5 n is

Step-by-step solution

Given f(x)=2x^3+mx^2-13x+n with roots x=2 and x=3 , we evaluate the polynomial at these values. Substituting x=2 gives 2(2)^3 + m(2)^2 - 13(2) + n = 0 , which simplifies to 4m + n = 10 . Substituting x=3 gives 2(3)^3 + m(3)^2 - 13(3) + n = 0 , yielding 9m + n = -15 . Subtracting the first equation from the second gives 5m = -25 , so m = -5 . Substituting into 4m + n = 10 gives 4(-5) + n = 10 , hence n = 30 . Finally, 4m + 5n = 4(-5) + 5(30) = -20 + 150 = 130 . The result is 130 .