Given that `x+2 and x-3` are factors of `x^(3) + ax +b`. Calculate the values of a and b. Also, find the remaining factor.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that \( x + 2 \) and \( x - 3 \) are factors of the polynomial \( P(x) = x^3 + ax + b \). We will also find the remaining factor.

### Step 1: Set up the equations using the factors
Since \( x + 2 \) is a factor, substituting \( x = -2 \) into \( P(x) \) should yield 0:
\[ P(-2) = (-2)^3 + a(-2) + b = 0 \]
This simplifies to:
\[ -8 - 2a + b = 0 \quad \text{(Equation 1)} \]

Since \( x - 3 \) is a factor, substituting \( x = 3 \) into \( P(x) \) should also yield 0:
\[ P(3) = 3^3 + a(3) + b = 0 \]
This simplifies to:
\[ 27 + 3a + b = 0 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations
We have the following two equations:
1. \( -2a + b = 8 \) (from Equation 1)
2. \( 3a + b = -27 \) (from Equation 2)

Now, we can eliminate \( b \) by subtracting Equation 1 from Equation 2:
\[ (3a + b) - (-2a + b) = -27 - 8 \]
This simplifies to:
\[ 5a = -35 \]
Thus, we find:
\[ a = -7 \]

### Step 3: Substitute \( a \) back to find \( b \)
Now, substitute \( a = -7 \) back into Equation 1:
\[ -2(-7) + b = 8 \]
This simplifies to:
\[ 14 + b = 8 \]
Thus,
\[ b = 8 - 14 = -6 \]

### Step 4: Write the polynomial with found values
Now we can write the polynomial:
\[ P(x) = x^3 - 7x - 6 \]

### Step 5: Find the remaining factor
We know that \( P(x) \) can be expressed as:
\[ P(x) = (x + 2)(x - 3)(\text{remaining factor}) \]
Let the remaining factor be \( Q(x) \). We can find it by dividing \( P(x) \) by \( (x + 2)(x - 3) \).

First, calculate \( (x + 2)(x - 3) \):
\[ (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]

Now, divide \( P(x) \) by \( x^2 - x - 6 \):
Using polynomial long division:
1. Divide \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 - x - 6 \) to get \( x^3 - x^2 - 6x \).
3. Subtract this from \( P(x) \):
\[ (x^3 - 7x - 6) - (x^3 - x^2 - 6x) = x^2 - x - 6 \]
4. Now, divide \( x^2 \) by \( x^2 \) to get \( 1 \).
5. Multiply \( 1 \) by \( x^2 - x - 6 \) to get \( x^2 - x - 6 \).
6. Subtract this from \( x^2 - x - 6 \) to get \( 0 \).

Thus, the remaining factor is:
\[ Q(x) = x + 1 \]

### Final Result
The values of \( a \) and \( b \) are:
- \( a = -7 \)
- \( b = -6 \)

The remaining factor is:
- \( x + 1 \)