If `f(x)=2x^3+mx^2-13x+n` and 2 and 3 are 2 roots of the equations f(x)=0, then values of m and n are

Options

A. `-5, -30`
B. `-5, 30`
C. 5, 30
D. none of these

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) in the polynomial function \( f(x) = 2x^3 + mx^2 - 13x + n \) given that 2 and 3 are roots of the equation \( f(x) = 0 \).

### Step 1: Use the fact that 2 is a root
Since 2 is a root, we can substitute \( x = 2 \) into the function and set it equal to zero:
\[ f(2) = 2(2)^3 + m(2)^2 - 13(2) + n = 0 \]
Calculating this gives:
\[ f(2) = 2(8) + m(4) - 26 + n = 0 \]
\[ 16 + 4m - 26 + n = 0 \]
\[ 4m + n - 10 = 0 \]
Thus, we can express this as our first equation:
\[ 4m + n = 10 \quad \text{(Equation 1)} \]

### Step 2: Use the fact that 3 is a root
Similarly, since 3 is also a root, we substitute \( x = 3 \) into the function:
\[ f(3) = 2(3)^3 + m(3)^2 - 13(3) + n = 0 \]
Calculating this gives:
\[ f(3) = 2(27) + m(9) - 39 + n = 0 \]
\[ 54 + 9m - 39 + n = 0 \]
\[ 9m + n + 15 = 0 \]
Thus, we can express this as our second equation:
\[ 9m + n = -15 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have the following system of equations:
1. \( 4m + n = 10 \)
2. \( 9m + n = -15 \)

We can eliminate \( n \) by subtracting Equation 1 from Equation 2:
\[ (9m + n) - (4m + n) = -15 - 10 \]
This simplifies to:
\[ 5m = -25 \]
Thus, we find:
\[ m = -5 \]

### Step 4: Substitute \( m \) back to find \( n \)
Now that we have \( m \), we can substitute it back into Equation 1 to find \( n \):
\[ 4(-5) + n = 10 \]
\[ -20 + n = 10 \]
\[ n = 30 \]

### Final Answer
The values of \( m \) and \( n \) are:
\[ m = -5, \quad n = 30 \]