If `(x +2) and (x - 3)` are the factors of the function `(x ^(3) + ax +b) ,` then find the values of a and b.

Options

A. `a =- 7, b =- 6`
B. `a =- 6, b =- 7`
C. `a =6, b =- 7`
D. `a =- 7, b = 6`

Answer

To find the values of \( a \) and \( b \) given that \( (x + 2) \) and \( (x - 3) \) are factors of the polynomial \( f(x) = x^3 + ax + b \), we can follow these steps:

### Step 1: Set up the equations using the factors
Since \( (x + 2) \) is a factor, we know that \( f(-2) = 0 \). Similarly, since \( (x - 3) \) is a factor, we have \( f(3) = 0 \).

### Step 2: Calculate \( f(-2) \)
Substituting \( x = -2 \) into the polynomial:
\[ f(-2) = (-2)^3 + a(-2) + b = -8 - 2a + b \]
Setting this equal to zero gives us our first equation:
\[ -8 - 2a + b = 0 \quad \text{(Equation 1)} \]

### Step 3: Calculate \( f(3) \)
Now substituting \( x = 3 \) into the polynomial:
\[ f(3) = 3^3 + a(3) + b = 27 + 3a + b \]
Setting this equal to zero gives us our second equation:
\[ 27 + 3a + b = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have the following system of equations:
1. \( -8 - 2a + b = 0 \)
2. \( 27 + 3a + b = 0 \)

We can rearrange Equation 1 to express \( b \):
\[ b = 8 + 2a \quad \text{(Substituting this into Equation 2)} \]

Substituting \( b \) into Equation 2:
\[ 27 + 3a + (8 + 2a) = 0 \]
This simplifies to:
\[ 27 + 3a + 8 + 2a = 0 \]
Combining like terms:
\[ 5a + 35 = 0 \]
Solving for \( a \):
\[ 5a = -35 \implies a = -7 \]

### Step 5: Substitute \( a \) back to find \( b \)
Now substitute \( a = -7 \) back into the equation for \( b \):
\[ b = 8 + 2(-7) = 8 - 14 = -6 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ \boxed{-7} \quad \text{and} \quad \boxed{-6} \]