Question

If x + 2  and x − 1 are the factors of x³ + 10x² + mx + n, then the values of m and n are respectively

Options

• 5 and −3

• 17 and −8

• 7 and −18

•  23 and −19

Answer 1

It is given (x + 2)and (x - 1)are the factors of the polynomial

`f(x) = x^3 + 10x^2 + mx + n`

i.e., f(-2) =0 and f(1) = 0

Now

`f(-2) = (-2)^3 + 10(-2)^2 + m(-2) + n =0`

\[- 8 + 40 - 2m + n = 0\]

\[ \Rightarrow - 2m + n = - 32\]

\[ \Rightarrow 2m - n = 32 . . . (1)\]

`f(1) = (1)^3 + 10(1)^2 +m(1) + n = 0`

`1 + 10 +m + n = 0`

`  m + n = -11      ........ (2)`

Solving equation (1) and (2) we get

m = 7 and n = − 18