Given that `x=-4` is a solution of `x^(3)-x^(2)-14x+24=0`. The other solutions are ________.

Options

A. 1,3
B. 2,3
C. 1,4
D. 2,5

Answer

To find the other solutions of the polynomial equation \(x^3 - x^2 - 14x + 24 = 0\) given that \(x = -4\) is one of the solutions, we can follow these steps:

### Step 1: Factor the Polynomial
Since \(x = -4\) is a solution, we can factor the polynomial using \(x + 4\) as one of the factors. We will perform polynomial long division to divide \(x^3 - x^2 - 14x + 24\) by \(x + 4\).

### Step 2: Perform Polynomial Long Division
1. Divide the leading term \(x^3\) by \(x\) to get \(x^2\).
2. Multiply \(x^2\) by \(x + 4\) to get \(x^3 + 4x^2\).
3. Subtract \(x^3 + 4x^2\) from \(x^3 - x^2\) to get \(-5x^2 - 14x\).
4. Bring down the next term (which is \(+24\)) to get \(-5x^2 - 14x + 24\).
5. Divide \(-5x^2\) by \(x\) to get \(-5x\).
6. Multiply \(-5x\) by \(x + 4\) to get \(-5x^2 - 20x\).
7. Subtract \(-5x^2 - 20x\) from \(-5x^2 - 14x + 24\) to get \(6x + 24\).
8. Divide \(6x\) by \(x\) to get \(6\).
9. Multiply \(6\) by \(x + 4\) to get \(6x + 24\).
10. Subtract \(6x + 24\) from \(6x + 24\) to get \(0\).

Thus, we have:

\[ x^3 - x^2 - 14x + 24 = (x + 4)(x^2 - 5x + 6) \]

### Step 3: Factor the Quadratic
Next, we need to factor the quadratic \(x^2 - 5x + 6\):

1. We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)).
2. The numbers \(-2\) and \(-3\) satisfy this condition.

Thus, we can factor the quadratic as:

\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 4: Write the Complete Factorization
Now we can write the complete factorization of the polynomial:

\[ x^3 - x^2 - 14x + 24 = (x + 4)(x - 2)(x - 3) \]

### Step 5: Find the Other Solutions
Setting each factor equal to zero gives us the solutions:

1. \(x + 4 = 0 \Rightarrow x = -4\) (given)
2. \(x - 2 = 0 \Rightarrow x = 2\)
3. \(x - 3 = 0 \Rightarrow x = 3\)

Thus, the other solutions are \(x = 2\) and \(x = 3\).

### Final Answer
The other solutions are **2 and 3**.

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