Find the relation between the roots and the coefficients of the cubic equation .
`3x^3-10x^2+7x+10=0`

Answer

To find the relation between the roots and the coefficients of the cubic equation \(3x^3 - 10x^2 + 7x + 10 = 0\), we will follow these steps:

### Step 1: Identify the coefficients
The given cubic equation is:

\[ 3x^3 - 10x^2 + 7x + 10 = 0 \]

We can compare this with the general form of a cubic equation:

\[ ax^3 + bx^2 + cx + d = 0 \]

From this comparison, we identify the coefficients:
- \(a = 3\)
- \(b = -10\)
- \(c = 7\)
- \(d = 10\)

### Step 2: Use Vieta's formulas
Vieta's formulas relate the coefficients of the polynomial to sums and products of its roots. Let the roots of the equation be \(\alpha\), \(\beta\), and \(\gamma\).

1. **Sum of the roots**:
\[ \alpha + \beta + \gamma = -\frac{b}{a} = -\frac{-10}{3} = \frac{10}{3} \]

2. **Sum of the products of the roots taken two at a time**:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{7}{3} \]

3. **Product of the roots**:
\[ \alpha\beta\gamma = -\frac{d}{a} = -\frac{10}{3} \]

### Step 3: Summarize the relations
From the above calculations, we have established the following relations between the roots and the coefficients of the cubic equation:

1. \(\alpha + \beta + \gamma = \frac{10}{3}\)
2. \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{7}{3}\)
3. \(\alpha\beta\gamma = -\frac{10}{3}\)

### Conclusion
The relations between the roots and the coefficients of the cubic equation \(3x^3 - 10x^2 + 7x + 10 = 0\) are as follows:

- The sum of the roots is \(\frac{10}{3}\).
- The sum of the products of the roots taken two at a time is \(\frac{7}{3}\).
- The product of the roots is \(-\frac{10}{3}\).

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