For what values of m and n Does the expression

( mx^4 + nx^3 + 14x^2 + 9x + 2)

divided by ( x^2 + 3x + 1 ) leaves a remainder ,
(-11x - 4 ) .. ? I divided it , nut the whole m and n remainder thingy made me so confused !!!

Try to use remainder therom reverse it

from divident subtract remainder then divide with no remainder ans will come

To find the values of m and n that satisfy the given condition, we need to use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by (x - a), the remainder is f(a).

In this case, we have to divide the polynomial ( mx^4 + nx^3 + 14x^2 + 9x + 2) by ( x^2 + 3x + 1 ). The remainder is given as (-11x - 4). So, we will equate the remainder to (-11x - 4) and solve for m and n.

Step 1: Divide the given polynomial by (x^2 + 3x + 1) using long division or polynomial division method:

___________________________________
x^2 + 3x + 1 | mx^4 + nx^3 + 14x^2 + 9x + 2
- (mx^4 + 3mx^3 + mx^2)
-------------------------------
0 + (n - 3m)x^3 + (14 - m)x^2 + (9 - m)x + 2

Step 2: The remainder obtained after division is (n - 3m)x^3 + (14 - m)x^2 + (9 - m)x + 2

Step 3: Equate the remainder to (-11x - 4):

(n - 3m)x^3 + (14 - m)x^2 + (9 - m)x + 2 = -11x - 4

Step 4: Now, compare the coefficients of x^3, x^2, x, and constant terms on both sides of the equation:

n - 3m = 0 ---- (1) (coefficient of x^3)
14 - m = -11 ---- (2) (coefficient of x^2)
9 - m = 0 ---- (3) (coefficient of x)
2 = -4 ---- (4) (constant term)

Step 5: Solve the system of equations (1), (2), (3), and (4) simultaneously to find the values of m and n.

From equation (1), we have n - 3m = 0, which implies n = 3m. Substitute this value of n into equation (2):

14 - m = -11
-m = -25
m = 25

Substitute the value of m = 25 into equation (3):

9 - m = 0
9 - 25 = 0
-16 = 0

Since -16 ≠ 0, there is no solution for n.

Therefore, the value of m is 25, and there is no solution for n that satisfies the given condition.