Question

Determine the values of m and n when 2x^3 -mx^2 + nx -2 is divided by x +1, the remainder is -12 and x-2 is a factor

Answer 1

Let f(x)  =  2x^3 -mx^2 + nx -2 ----------> (1)

if (x - 2) is factor of f(x) then f(2) = 0

Substitute x = 2 in f(x)

f( 2 )  =  2(2)^3 -m(2)^2 + n(2) -2

0  =  2(8) -m(4) + 2n -2                                   [ Since f(2)  =  0 ]

0  =  16 - 4m +2n -2

4m - 2n  =  14----------------------> (2)

if f(x) is devisable by (x + 1) then f(-1) = R     [ R stands for remainder ]

Substitute x = -1 in f(x)

f( -1 )  =  2(-1)^3 -m(-1)^2 + n(-1) -2

-12  =   2(-1) - m(1) - n - 2                              [ Since f(-1)  =   R  =  -12 ]

 

 

-12  =   -2 - m - n - 2

m + n  =  -4 + 12

m + n  = 8 -------------------> (3)

 

Multiply Eq (3) with 2 and to Eq (2)

Eq (3) X 2 =====>  2m + 2n  =  16

Eq (2) ========>  4m - 2n  = 14

                           ________________

                                6m + 0  =  30

                                     6m   =  30

                                       m   =  30/6

                                       m   =  5

Substitute m = 5 in Eq (3)

5 + n  =  8

n  =  8 - 5

n  =  3.

 

Answer :

Solution is m = 5 and n  =  3..