If `x=-4` is a root of a `Delta=|(x,2,3),(1,x,1),(3,2,x)|=0`, then find the other two roots.

Answer

To solve the problem, we need to find the roots of the determinant equation given that one root is \( x = -4 \). The determinant is defined as:

\[ \Delta = \begin{vmatrix} x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x \end{vmatrix} \]

We need to set this determinant equal to zero and find the other roots.

### Step 1: Calculate the Determinant

We will expand the determinant using the first row:

\[ \Delta = x \begin{vmatrix} x & 1 \\ 2 & x \end{vmatrix} - 2 \begin{vmatrix} 1 & 1 \\ 3 & x \end{vmatrix} + 3 \begin{vmatrix} 1 & x \\ 3 & 2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \(\begin{vmatrix} x & 1 \\ 2 & x \end{vmatrix} = x^2 - 2\)

2. \(\begin{vmatrix} 1 & 1 \\ 3 & x \end{vmatrix} = x - 3\)

3. \(\begin{vmatrix} 1 & x \\ 3 & 2 \end{vmatrix} = 2 - 3x\)

Now substituting these values back into the determinant:

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

### Step 2: Simplify the Expression

Now we simplify the expression:

\[ \Delta = x^3 - 2x - 2x + 6 + 6 - 9x \]

Combining like terms:

\[ \Delta = x^3 - 13x + 12 \]

### Step 3: Set the Determinant to Zero

Now we set the determinant equal to zero:

\[ x^3 - 13x + 12 = 0 \]

### Step 4: Factor the Polynomial

Since we know that \( x = -4 \) is a root, we can factor \( x + 4 \) out of the polynomial. We will perform polynomial long division:

1. Divide \( x^3 - 13x + 12 \) by \( x + 4 \).

- \( x^3 \div x = x^2 \)
- Multiply \( x^2 \) by \( x + 4 \) gives \( x^3 + 4x^2 \)
- Subtract: \( -4x^2 - 13x + 12 \)
- Divide \( -4x^2 \div x = -4x \)
- Multiply: \( -4x(x + 4) = -4x^2 - 16x \)
- Subtract: \( 3x + 12 \)
- Divide \( 3x \div x = 3 \)
- Multiply: \( 3(x + 4) = 3x + 12 \)
- Subtract: \( 0 \)

Thus, we have:

\[ x^3 - 13x + 12 = (x + 4)(x^2 - 4x + 3) \]

### Step 5: Factor the Quadratic

Now we factor the quadratic \( x^2 - 4x + 3 \):

\[ x^2 - 4x + 3 = (x - 3)(x - 1) \]

### Step 6: Write the Complete Factorization

Putting it all together, we have:

\[ (x + 4)(x - 3)(x - 1) = 0 \]

### Step 7: Find the Roots

The roots of the equation are:

1. \( x = -4 \)
2. \( x = 3 \)
3. \( x = 1 \)

### Final Answer

The other two roots are \( x = 3 \) and \( x = 1 \).

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