If (x+4) is a factor of the polynomial `x^(3)-x^(2)-14x+24`, find the other factors.

Answer

To find the other factors of the polynomial \( p(x) = x^3 - x^2 - 14x + 24 \) given that \( (x + 4) \) is a factor, we can follow these steps:

### Step 1: Polynomial Division
Since \( (x + 4) \) is a factor of \( p(x) \), we can divide \( p(x) \) by \( (x + 4) \) using polynomial long division.

1. **Divide the leading term**: The leading term of \( p(x) \) is \( x^3 \) and the leading term of \( (x + 4) \) is \( x \). Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. **Multiply**: Multiply \( (x + 4) \) by \( x^2 \) to get \( x^3 + 4x^2 \).
3. **Subtract**: Subtract \( (x^3 + 4x^2) \) from \( p(x) \):
\[ (x^3 - x^2 - 14x + 24) - (x^3 + 4x^2) = -5x^2 - 14x + 24 \]

### Step 2: Repeat the Division
Now, we continue the division with the new polynomial \( -5x^2 - 14x + 24 \).

1. **Divide the leading term**: The leading term is \( -5x^2 \). Divide by \( x \) to get \( -5x \).
2. **Multiply**: Multiply \( (x + 4) \) by \( -5x \) to get \( -5x^2 - 20x \).
3. **Subtract**: Subtract \( (-5x^2 - 20x) \) from \( -5x^2 - 14x + 24 \):
\[ (-5x^2 - 14x + 24) - (-5x^2 - 20x) = 6x + 24 \]

### Step 3: Final Division
Now, we continue with \( 6x + 24 \).

1. **Divide the leading term**: The leading term is \( 6x \). Divide by \( x \) to get \( 6 \).
2. **Multiply**: Multiply \( (x + 4) \) by \( 6 \) to get \( 6x + 24 \).
3. **Subtract**: Subtract \( (6x + 24) \) from \( 6x + 24 \):
\[ (6x + 24) - (6x + 24) = 0 \]

### Step 4: Result of Division
The result of the division is:
\[ p(x) = (x + 4)(x^2 - 5x + 6) \]

### Step 5: Factor the Quadratic
Now we need to factor the quadratic \( x^2 - 5x + 6 \).

1. **Find two numbers that multiply to \( 6 \) and add to \( -5 \)**: The numbers are \( -2 \) and \( -3 \).
2. **Write the factors**: Thus, we can factor it as:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 6: Final Factors
Putting it all together, we have:
\[ p(x) = (x + 4)(x - 2)(x - 3) \]

### Conclusion
The other factors of the polynomial \( x^3 - x^2 - 14x + 24 \) are \( (x - 2) \) and \( (x - 3) \).

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