Question

find the value of m and n so that the polynomial (x) =
${x}^{3} -6 {x}^{2} + mx - n$
and is divisible by (x-1) and (x-2)
I want solutions....Answer is m=1 , n=6

Answer 1

Given that
(x-1) & (x-2) is factor of equation which is divide the equation

Now put the value in equation
X = 1

${1}^{3} - 6( {1})^{2} + m - n = 0 \\ m - n = 5.........(1) \\ \\ now \: put \: the \: value \: of \: \\ x = 2 \\ \\ {2}^{3} - 6( {2})^{2} + 2m - n = 0 \\ 8 - 24 + 2m - n \\ 2m - n = 16 \\ \\ now \: solve \: eqution \: (1) \: and \: (2) \\ m - n = 5 \\ 2m - n = 16 \: \: \: mutiply \: in \: eqution \: (1) \: by \: 2 \: and \: \\ eqution \: (2) \: by \: 1 \\ \\ now \: we \: get \\ 2m - 2n = 10 \\ 2m - n = 16 \\ \frac{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ - n = - 6 } \\ \: \\ n = 6$
m-n = 5
m-6 =5
m= 5+6
m= 11 Ans

Hope it's helpful ✌
!