Solve `18x^3+81x^2+121x+60=0` given that one roots is equal to half the sum of the remainging roots .

Answer

To solve the equation \( 18x^3 + 81x^2 + 121x + 60 = 0 \) given that one root is equal to half the sum of the remaining roots, we can follow these steps:

### Step 1: Define the Roots
Let the roots of the polynomial be \( \alpha - \beta, \alpha, \alpha + \beta \). According to the problem, one root (\( \alpha \)) is equal to half the sum of the remaining roots:
\[ \alpha = \frac{(\alpha - \beta) + (\alpha + \beta)}{2} \]
This simplifies to:
\[ \alpha = \frac{2\alpha}{2} \implies \alpha = \alpha \]
This confirms our assumption about the roots.

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we know:
1. The sum of the roots \( (\alpha - \beta) + \alpha + (\alpha + \beta) = 3\alpha \) is equal to \( -\frac{b}{a} = -\frac{81}{18} = -\frac{9}{2} \).
2. Therefore, we have:
\[ 3\alpha = -\frac{9}{2} \implies \alpha = -\frac{3}{2} \]

### Step 3: Identify the Remaining Roots
Now we can find the other roots:
- The first root is \( \alpha - \beta = -\frac{3}{2} - \beta \)
- The second root is \( \alpha = -\frac{3}{2} \)
- The third root is \( \alpha + \beta = -\frac{3}{2} + \beta \)

### Step 4: Sum of the Roots
The sum of the roots can be expressed as:
\[ (-\frac{3}{2} - \beta) + (-\frac{3}{2}) + (-\frac{3}{2} + \beta) = -\frac{9}{2} \]

### Step 5: Product of the Roots
Using Vieta's formulas again, the product of the roots is given by:
\[ (\alpha - \beta) \cdot \alpha \cdot (\alpha + \beta) = -\frac{c}{a} = -\frac{121}{18} \]
Substituting \( \alpha = -\frac{3}{2} \):
\[ (-\frac{3}{2} - \beta)(-\frac{3}{2})(-\frac{3}{2} + \beta) = -\frac{121}{18} \]

### Step 6: Polynomial Division
Now we can divide the polynomial \( 18x^3 + 81x^2 + 121x + 60 \) by \( x + \frac{3}{2} \) (or \( 2x + 3 \) after multiplying by 2 to eliminate the fraction).

Perform polynomial long division:
1. Divide \( 18x^3 \) by \( 2x \) to get \( 9x^2 \).
2. Multiply \( 9x^2 \) by \( 2x + 3 \) and subtract from the original polynomial.
3. Repeat the process for the resulting polynomial until you reach a quadratic.

### Step 7: Solve the Quadratic
After division, you will have a quadratic equation. Use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
to find the other two roots.

### Final Roots
The roots of the polynomial are:
1. \( -\frac{3}{2} \)
2. \( -\frac{4}{3} \)
3. \( -\frac{5}{3} \)