Let `a_1,a_2,a_3` be the roots of the polynomial `x^3+x^2+3x+1`.Find the value of `a_1+a_2+a_3+a_1a_2+a_2a_3+a_3a_1+a_1a_2a_3`

Options

A. 5
B. 6
C. 3
D. 1

Answer

To find the value of \( a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_3a_1 + a_1a_2a_3 \) for the polynomial \( x^3 + x^2 + 3x + 1 \), we can use Vieta's formulas, which relate the coefficients of the polynomial to sums and products of its roots.

### Step 1: Identify the coefficients
The polynomial is given as:
\[ x^3 + x^2 + 3x + 1 \]
Here, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^3 \))
- \( b = 1 \) (coefficient of \( x^2 \))
- \( c = 3 \) (coefficient of \( x \))
- \( d = 1 \) (constant term)

### Step 2: Calculate \( a_1 + a_2 + a_3 \)
According to Vieta's formulas, the sum of the roots \( a_1 + a_2 + a_3 \) is given by:
\[ a_1 + a_2 + a_3 = -\frac{b}{a} = -\frac{1}{1} = -1 \]

### Step 3: Calculate \( a_1a_2 + a_2a_3 + a_3a_1 \)
The sum of the products of the roots taken two at a time is given by:
\[ a_1a_2 + a_2a_3 + a_3a_1 = \frac{c}{a} = \frac{3}{1} = 3 \]

### Step 4: Calculate \( a_1a_2a_3 \)
The product of the roots is given by:
\[ a_1a_2a_3 = -\frac{d}{a} = -\frac{1}{1} = -1 \]

### Step 5: Combine the results
Now we can substitute these values into the expression we need to evaluate:
\[ a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_3a_1 + a_1a_2a_3 = (-1) + 3 + (-1) \]
Calculating this gives:
\[ = -1 + 3 - 1 = 1 \]

### Final Result
Thus, the value of \( a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_3a_1 + a_1a_2a_3 \) is:
\[ \boxed{1} \]