If `(x+2) and (x-1) ` are factors of `(x^3+10x^2+mx+n)` then

Options

A. `m=5,n=-3`
B. `m=7,n=-18`
C. `m=17,n=-8`
D. `m=23,n=-19`

Explanation

To solve the problem where `(x + 2)` and `(x - 1)` are factors of the polynomial `(x^3 + 10x^2 + mx + n)`, we can follow these steps:

### Step-by-Step Solution:

1. **Use the Factor Theorem**: Since `(x + 2)` is a factor, substituting `x = -2` into the polynomial should yield zero.
\[ P(-2) = (-2)^3 + 10(-2)^2 + m(-2) + n = 0 \]
Simplifying this gives:
\[ -8 + 40 - 2m + n = 0 \]
\[ 32 - 2m + n = 0 \quad \text{(Equation 1)} \]

2. **Substituting for the second factor**: Now, since `(x - 1)` is also a factor, substituting `x = 1` into the polynomial should also yield zero.
\[ P(1) = (1)^3 + 10(1)^2 + m(1) + n = 0 \]
Simplifying this gives:
\[ 1 + 10 + m + n = 0 \]
\[ 11 + m + n = 0 \quad \text{(Equation 2)} \]

3. **Solving the equations**: Now we have two equations:
- From Equation 1: \( n = 2m - 32 \)
- From Equation 2: \( m + n = -11 \)

Substitute the expression for \( n \) from Equation 1 into Equation 2:
\[ m + (2m - 32) = -11 \]
Simplifying this gives:
\[ 3m - 32 = -11 \]
\[ 3m = 21 \]
\[ m = 7 \]

4. **Finding n**: Now substitute \( m = 7 \) back into Equation 1 to find \( n \):
\[ n = 2(7) - 32 \]
\[ n = 14 - 32 \]
\[ n = -18 \]

5. **Final Result**: The values of \( m \) and \( n \) are:
\[ m = 7, \quad n = -18 \]