When mx^3-nx^2+5x-1 is divided by x+2 the remainder is -39.

Question

when divided by x-1 the remainder is 3. find n and m.?  

Answer 1

The polynomial is f(x) = mx³ - nx²  + 5x - 1

From remainder theorem,

When you divide a polynomial f(x) by x - a the remainder r will be f(a).

x + 2 is a divisor.

f(-2) = m(-2)³ - n(-2)²  + 5(-2) - 1 = - 39

- 8m - 4n  - 10 - 1 = - 39

- 8m - 4n = - 39 + 11

- 8m - 4n = - 28 ---> (1)

x - 1 is a divisor.

f(1) = m(1)³ - n(1)²  + 5(1) - 1 =  3

m - n + 5 - 1 = 3

m - n = - 1  ---> (2)

Solve the equations (1) & (2).

From (2), m = - 1 + n ---> (3)

Substitute the value of m in (1).

- 8( -1 + n) - 4n = - 28

8 - 8n - 4n = - 28

- 12n = - 28 - 8

- 12n = -36

n =  3

Substitute the value of m in (3).

m = - 1 + 3

m = 2

Solution m = 2 and n = 3.