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

Given that while deviding f(x) with ( x + 1) Then remainder is - 12.

Hence, f( -1 ) = - 12

Substittute x = -1 in f(x)

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

-12   =  2(-1) - m(1) - n - 2

-12   =  -2 - m - n - 2

m + n  =  12 - 2 - 2

m + n  =  8 ------------------> (1)

And also given that (x - 2) is a factor of f(x)

Hence, f(2)  =  0

f(2)  =  2(2)^3 - m(2)^2 + n(2) - 2

0  =  2(8) - m(4) + 2n - 2

0  =  16 - 4m + 2n - 2

0  =  14 - 4m + 2n

4m - 2n  =  14

2(2m - n)  =  14

(2m - n)  =  14/2

2m - n  =  7 -------------------> (2)

Add both the equations (1) and (2)

    m + n  =  8 

  2m - n  =  7

-------------------

   3m  =  15

m  =  15/3

m  =  5

Substitute m = 5 in Eq(1)

5 + n  =  8

n  =  8 - 5

n  =  3

Answer :

The solutions are m = 5 and n = 3.