If `alpha, beta` and 1 are the roots of `x^3-2x^2-5x+6=0 ` , then find `alpha and beta`

Answer

To solve the problem, we need to find the values of `α` and `β` given that `1`, `α`, and `β` are the roots of the polynomial equation \( x^3 - 2x^2 - 5x + 6 = 0 \).

### Step-by-Step Solution:

1. **Identify the given polynomial:**
The polynomial is \( x^3 - 2x^2 - 5x + 6 = 0 \).

2. **Use Vieta's Formulas:**
According to Vieta's formulas, for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) with roots \( r_1, r_2, r_3 \):
- The sum of the roots \( r_1 + r_2 + r_3 = -\frac{b}{a} \)
- The sum of the products of the roots taken two at a time \( r_1r_2 + r_2r_3 + r_3r_1 = \frac{c}{a} \)
- The product of the roots \( r_1r_2r_3 = -\frac{d}{a} \)

Here, \( a = 1, b = -2, c = -5, d = 6 \).

3. **Calculate the sum of the roots:**
Given that the roots are \( 1, α, β \):
\[ 1 + α + β = -\frac{-2}{1} = 2 \]
Thus, we can express this as:
\[ α + β = 2 - 1 = 1 \quad \text{(Equation 1)} \]

4. **Calculate the sum of the products of the roots taken two at a time:**
\[ 1 \cdot α + 1 \cdot β + α \cdot β = -5 \]
This simplifies to:
\[ α + β + αβ = -5 \]
Substituting \( α + β = 1 \) from Equation 1:
\[ 1 + αβ = -5 \]
Therefore:
\[ αβ = -5 - 1 = -6 \quad \text{(Equation 2)} \]

5. **Now we have a system of equations:**
From Equation 1:
\[ α + β = 1 \]
From Equation 2:
\[ αβ = -6 \]

6. **Use these equations to form a quadratic equation:**
The roots \( α \) and \( β \) can be found using the quadratic equation:
\[ x^2 - (α + β)x + αβ = 0 \]
Substituting the values:
\[ x^2 - 1x - 6 = 0 \]

7. **Solve the quadratic equation:**
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 + 24}}{2} \]
\[ x = \frac{1 \pm \sqrt{25}}{2} \]
\[ x = \frac{1 \pm 5}{2} \]

8. **Calculate the roots:**
This gives us two possible solutions:
\[ x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{-4}{2} = -2 \]

9. **Final values of α and β:**
Therefore, the values of \( α \) and \( β \) are:
\[ α = 3, \quad β = -2 \]

### Summary:
The roots \( α \) and \( β \) are \( 3 \) and \( -2 \) respectively.