Expert: Sherry Wallin - 6/30/2011

Question
if 2x^3+Mx^2+Nx+6 is divisible ny both x-1 & x+2 , find M & N.

Answer
Jasmin~

Do you know how to do synthetic division? If both x-1 and x+2 divide your polynomial that means they have a zero remainder. Once you do the synthetic division your remainders will need to be set to 0 and you will solve your two equations with two unknowns, namely M and N.

if x-1 is a factor of 2x^3+Mx^2+Nx + 6 then your synthetic division looks like this:

1 |____2____________M______________N______________6_______________

    _________________2_____________M+2_________M+N+2____________
          2                      M+2                     M+N+2                 M+N+8 --->  this remainder needs to be zero since x-1 is a factor

-2 |____2____________M______________N______________6_______________

      ________________-4___________-2M+8_________4M-2N-16____________
             2                      M-4                 -2M+N+8                 4M-2N-10 --->  this remainder needs to be zero since x+2 is a factor

  M + N + 8 = 0   multiply by 2 and add to second equation
4M - 2N -10 = 0
_______________

2M + 2N +16 = 0
4M -  2N  - 10 = 0
__________________
6M + 6 = 0 => 6M = -6 => M = -1

now substitute M = -1  into M + N + 8 = 0 => -1 + N + 8 = 0 => N = -7

You now have the values for both M and N. Redo the long/synthetic division with these values to verify that they work!!

Math Prof