Question

any polynomial expert

Answer 1

HEY USER !!

HERE IS UR ANSWER ⤵⤵

➡Given ,

$f(x) = x ^3 + mx ^2 + nx + 6$

$g(x) = x - 2$

Since, x-2 is a factor of f(x) , so,

$x - 2 = 0$

$x = 2$

$f(2) = 2 ^3 + m*2 ^2 + n*2 + 6$

$=> 8 + 4m + 2n + 6 = 0$

$=> 14 + 4m + 2n = 0$

$=> 4m + 2n = -14$

$=> 2(2m + n) = -14$

$=> 2m + n = -14/2$

$=> n = -7 - 2m ......(i)$

When f(x) is divided by x-3 , leaves a remainder 3 ,

$x - 3 = 0$

$x = 3$

$f(3) = 3 ^3 + m*3 ^2 + n*3 + 6$

$=> 27 + 9m + 3n + 6 = 3$

$=> 33 + 9m + 3n - 3 = 0$

$=> 9m + 3n = -30$

$=> 3(3m + n) = -30$

$=> 3m + n = -30/3$

$=> 3m + n = -10......(ii)$

Now , put the value of n as -7 - 2m in equation (ii).

$3m + n = -10$

$3m + (-7) - 2m = -10$

$3m - 7 - 2m = -10$

$m - 7 = -10$

$m = -10 + 7$

$m = - 3$

$Now , n = -7 - 2m$

$= - 7 - 2*(-3)$

$= -7 + 6$

$= - 1$

✔$Therefore , m = -3 , n = -1$

HOPE IT HELPS....☺✌

Answer 2

HEYA !!

Here is ur solution.....

f(x) = x ^3 + mx ^2 + nx + 6
g(x) = x - 2

x - 2 = 0
x = 2

f(2) = 2 ^3 + m*2 ^2 + n*2 + 6

=> 8 + 4m + 2n + 6 = 0
=> 14 + 4m + 2n = 0
=> 4m + 2n = -14
=> 2(2m + n) = -14
=> 2m + n = -14/2
=> n = -7 - 2m (i)

Now,

x - 3 = 0
x = 3

f(3) = 3 ^3 + m*3 ^2 + n*3 + 6

=> 27 + 9m + 3n + 6 = 3
=> 33 + 9m + 3n - 3 = 0
=> 9m + 3n = -30
=> 3(3m + n) = -30
=> 3m + n = -30/3
=> 3m + n = -10 (ii)

Now ..

3m + n = -10
3m + (-7) - 2m = -10
3m - 7 - 2m = -10
m - 7 = -10
m = -10 + 7
m = - 3

Now , n = -7 - 2m

= - 7 - 2*(-3)
= -7 + 6
= - 1

So, m = -3 , n = -1

hope it help u..... :)