If `8x^(4) - 2x^(3) - 27x^(2) + 6x + 9 = 0` then `s_(1), s_(2), s_(3) , s_(4)` are

Answer

To solve the equation \( 8x^4 - 2x^3 - 27x^2 + 6x + 9 = 0 \) and find the values of \( s_1, s_2, s_3, \) and \( s_4 \), we will follow these steps:

### Step 1: Identify Coefficients
The given polynomial can be compared to the general form of a polynomial equation:
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
From the equation \( 8x^4 - 2x^3 - 27x^2 + 6x + 9 = 0 \), we identify the coefficients:
- \( a = 8 \)
- \( b = -2 \)
- \( c = -27 \)
- \( d = 6 \)
- \( e = 9 \)

### Step 2: Calculate \( s_1 \)
The value \( s_1 \) is the sum of the roots of the polynomial, given by the formula:
\[ s_1 = \alpha + \beta + \gamma + \delta = -\frac{b}{a} \]
Substituting the values:
\[ s_1 = -\frac{-2}{8} = \frac{2}{8} = \frac{1}{4} \]

### Step 3: Calculate \( s_2 \)
The value \( s_2 \) is the sum of the products of the roots taken two at a time:
\[ s_2 = \alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = \frac{c}{a} \]
Substituting the values:
\[ s_2 = \frac{-27}{8} \]

### Step 4: Calculate \( s_3 \)
The value \( s_3 \) is the sum of the products of the roots taken three at a time:
\[ s_3 = \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -\frac{d}{a} \]
Substituting the values:
\[ s_3 = -\frac{6}{8} = -\frac{3}{4} \]

### Step 5: Calculate \( s_4 \)
The value \( s_4 \) is the product of the roots:
\[ s_4 = \alpha\beta\gamma\delta = \frac{e}{a} \]
Substituting the values:
\[ s_4 = \frac{9}{8} \]

### Final Results
Thus, we have:
- \( s_1 = \frac{1}{4} \)
- \( s_2 = -\frac{27}{8} \)
- \( s_3 = -\frac{3}{4} \)
- \( s_4 = \frac{9}{8} \)