If ` 1,1,alpha` are the roots of ` x^3 -6x^2 +9 x-4=0` then find `' alpha'`

Answer

To find the value of \( \alpha \) given that \( 1, 1, \alpha \) are the roots of the polynomial equation \( x^3 - 6x^2 + 9x - 4 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The polynomial can be expressed in the standard form \( ax^3 + bx^2 + cx + d = 0 \). Here, we have:
- \( a = 1 \) (coefficient of \( x^3 \))
- \( b = -6 \) (coefficient of \( x^2 \))
- \( c = 9 \) (coefficient of \( x \))
- \( d = -4 \) (constant term)

### Step 2: Use the relationship for the sum of the roots
For a cubic equation, the sum of the roots is given by the formula:
\[ \text{Sum of roots} = -\frac{b}{a} \]
Substituting the values of \( b \) and \( a \):
\[ \text{Sum of roots} = -\frac{-6}{1} = 6 \]

### Step 3: Set up the equation for the sum of the roots
Since the roots are \( 1, 1, \alpha \), we can write:
\[ 1 + 1 + \alpha = 6 \]

### Step 4: Solve for \( \alpha \)
Now, simplify the equation:
\[ 2 + \alpha = 6 \]
Subtract 2 from both sides:
\[ \alpha = 6 - 2 \]
\[ \alpha = 4 \]

### Conclusion
The value of \( \alpha \) is \( 4 \).