If `x=-4` is a root of `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 evaluate the determinant and find the roots of the equation given that one root is \( x = -4 \).

### Step 1: Set up the determinant
We have the determinant:
\[ \Delta = \begin{vmatrix} x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x \end{vmatrix} \]
We need to find the value of this determinant and set it equal to zero.

### Step 2: Expand the determinant
We can 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 the 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. \(\begin{vmatrix} 1 & x \\ 3 & 2 \end{vmatrix} = 2 - 3x\)

Substituting these back into the determinant:
\[ \Delta = x(x^2 - 2) - 2(x - 3) + 3(2 - 3x) \]

### Step 3: Simplify the expression
Now, we simplify:
\[ \Delta = x^3 - 2x - 2x + 6 + 6 - 9x \]
Combining like terms:
\[ \Delta = x^3 - 13x + 12 \]

### Step 4: Set the determinant to zero
Since we know that \( x = -4 \) is a root, we substitute \( x = -4 \) into the equation:
\[ (-4)^3 - 13(-4) + 12 = 0 \]
Calculating this:
\[ -64 + 52 + 12 = 0 \]
This confirms that \( x = -4 \) is indeed a root.

### Step 5: Factor the polynomial
Now, we can factor the polynomial \( x^3 - 13x + 12 \) using synthetic division or polynomial long division by \( x + 4 \):
\[ x^3 - 13x + 12 = (x + 4)(x^2 - 4x + 3) \]

### Step 6: Factor the quadratic
Next, we factor the quadratic \( x^2 - 4x + 3 \):
\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]

### Step 7: Write the complete factorization
Thus, the complete factorization of the polynomial is:
\[ (x + 4)(x - 1)(x - 3) = 0 \]

### Step 8: Find the other roots
Setting each factor to zero gives us the roots:
1. \( x + 4 = 0 \) → \( x = -4 \) (given root)
2. \( x - 1 = 0 \) → \( x = 1 \) (other root)
3. \( x - 3 = 0 \) → \( x = 3 \) (other root)

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

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