Find the equation whose roots are
3 times the roots of ` 6x^4 -7x^3 +8x^2 -7x +2=0`

Answer

To find the equation whose roots are three times the roots of the polynomial \(6x^4 - 7x^3 + 8x^2 - 7x + 2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients of the given polynomial
The given polynomial is:
\[ 6x^4 - 7x^3 + 8x^2 - 7x + 2 = 0 \]
From this, we can identify the coefficients:
- \(A = 6\)
- \(B = -7\)
- \(C = 8\)
- \(D = -7\)
- \(E = 2\)

### Step 2: Use Vieta's formulas to find relationships between the roots
Let the roots of the polynomial be \(\alpha, \beta, \gamma, \delta\). According to Vieta's formulas:
1. The sum of the roots:
\[ \alpha + \beta + \gamma + \delta = -\frac{B}{A} = -\frac{-7}{6} = \frac{7}{6} \]

2. The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = \frac{C}{A} = \frac{8}{6} = \frac{4}{3} \]

3. The sum of the products of the roots taken three at a time:
\[ \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -\frac{D}{A} = -\frac{-7}{6} = \frac{7}{6} \]

4. The product of the roots:
\[ \alpha\beta\gamma\delta = \frac{E}{A} = \frac{2}{6} = \frac{1}{3} \]

### Step 3: Find the new roots
The new roots are \(3\alpha, 3\beta, 3\gamma, 3\delta\). We can find the relationships for the new roots:
1. The sum of the new roots:
\[ 3(\alpha + \beta + \gamma + \delta) = 3 \cdot \frac{7}{6} = \frac{7}{2} \]

2. The sum of the products of the new roots taken two at a time:
\[ 9(\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta) = 9 \cdot \frac{4}{3} = 12 \]

3. The sum of the products of the new roots taken three at a time:
\[ 27(\alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta) = 27 \cdot \frac{7}{6} = \frac{189}{6} = \frac{63}{2} \]

4. The product of the new roots:
\[ 81(\alpha\beta\gamma\delta) = 81 \cdot \frac{1}{3} = 27 \]

### Step 4: Form the new polynomial
Using the relationships derived from the new roots, we can form the polynomial:
\[ x^4 - \left(\frac{7}{2}\right)x^3 + 12x^2 - \left(\frac{63}{2}\right)x + 27 = 0 \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply the entire equation by 2:
\[ 2x^4 - 7x^3 + 24x^2 - 63x + 54 = 0 \]

### Final Answer
The required polynomial whose roots are three times the roots of the original polynomial is:
\[ \boxed{2x^4 - 7x^3 + 24x^2 - 63x + 54 = 0} \]