Suppose
`P(x)=|(x,-51,-71),(51,x,-73),(71,73,x)|`
Product of zeros of P(x) is

Options

A. 0
B. 195
C. `-195`
D. `-26433`

Answer

To find the product of the zeros of the polynomial \( P(x) = |(x, -51, -71), (51, x, -73), (71, 73, x)| \), we will follow these steps:

### Step 1: Calculate the Determinant
First, we need to compute the determinant of the matrix:

\[ P(x) = \begin{vmatrix} x & -51 & -71 \\ 51 & x & -73 \\ 71 & 73 & x \end{vmatrix} \]

### Step 2: Expand the Determinant
Using the formula for the determinant of a 3x3 matrix, we expand it:

\[ P(x) = x \begin{vmatrix} x & -73 \\ 73 & x \end{vmatrix} - (-51) \begin{vmatrix} 51 & -73 \\ 71 & x \end{vmatrix} - (-71) \begin{vmatrix} 51 & x \\ 71 & 73 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} x & -73 \\ 73 & x \end{vmatrix} = x^2 - (-73)(73) = x^2 + 5329 \)
2. \( \begin{vmatrix} 51 & -73 \\ 71 & x \end{vmatrix} = 51x + 73(71) = 51x + 5183 \)
3. \( \begin{vmatrix} 51 & x \\ 71 & 73 \end{vmatrix} = 51 \cdot 73 - 71 \cdot x = 3723 - 71x \)

Substituting these back into the determinant:

\[ P(x) = x(x^2 + 5329) + 51(51x + 5183) + 71(3723 - 71x) \]

### Step 3: Simplify the Expression
Now, we simplify \( P(x) \):

\[ P(x) = x^3 + 5329x + 51(51x + 5183) + 71(3723 - 71x) \]

Calculating the constants:

- \( 51(51x + 5183) = 2601x + 264933 \)
- \( 71(3723 - 71x) = 264753 - 5041x \)

Combining all terms:

\[ P(x) = x^3 + (5329 + 2601 - 5041)x + (264933 + 264753) \]

### Step 4: Collect Like Terms
Now, we collect like terms:

\[ P(x) = x^3 + 2889x + 529686 \]

### Step 5: Identify Coefficients
From the polynomial \( P(x) = x^3 + 2889x + 529686 \), we identify:

- \( A = 1 \) (coefficient of \( x^3 \))
- \( D = 529686 \) (constant term)

### Step 6: Calculate the Product of the Zeros
The product of the zeros of a polynomial \( ax^3 + bx^2 + cx + d \) is given by \( -\frac{D}{A} \).

Thus, the product of the zeros is:

\[ -\frac{529686}{1} = -529686 \]

### Final Answer
The product of the zeros of \( P(x) \) is \( -529686 \).

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