Question

find the value of m and n so that ( X - 1 ) and ( x + 2 ) are the factor of x ^3 + 10 X^2 + MX + n

Answer 1

Hiii friend,

(X-1) and (X+2) are the two factor of the given polynomial.

(X-1) = 0 OR (X+2) = 0

X = 1 OR X = -2

P(X) = X³+10X²+MX+N

P(1) = (1)³ + 10 × (1)² + M × 1 + N

=> 1 + 10 + M + N = 0

=> M+ N + 11 = 0

=> M + N = -11------------(1)

Also X = -2

P(X) = X³+10X²+MX+N

P(-2) = (-2)³ + 10 × (-2)² + M × -2 + N

=> -8 + 10 × 4 - 2M + N = 0

=> -8 + 40 - 2M + N = 0

=> -2M + N + 32 = 0

=> 2M - N = 32 ----------(2)

Now , we have two equations.

M + N = -11 and 2M -N = 32

From equation (1) we get,

M + N = -11

M = -11 - N ----------(3)

Putting the value of M in equation (2)

2M - N = 32

2(-11 - N) - N = 32

-22 - 2N -N = 32

-3N-22 = 32

-2N = 32+22

-2N = 54

N = -54/2 => -18

Putting N = -18 in equation (3)

M = -11- N => -11 - (-18) = -11 + 18 = 7


Hence,

N = -18 and M = 7

HOPE IT WILL HELP YOU.... :-)