Verify that 1,2,3 are the zeroes of the cubic polynomial
p(x)`=x^(3)-6x^(2)+11x-6` and verify the relation between its zeroes and coefficients.

Explanation

To verify that 1, 2, and 3 are the zeroes of the cubic polynomial \( p(x) = x^3 - 6x^2 + 11x - 6 \) and to check the relation between its zeroes and coefficients, we will follow these steps:

### Step 1: Verify the Zeroes

1. **Check for \( x = 1 \)**:
\[ p(1) = 1^3 - 6 \cdot 1^2 + 11 \cdot 1 - 6 \]
\[ = 1 - 6 + 11 - 6 = 0 \]
Hence, \( x = 1 \) is a zero of the polynomial.

2. **Check for \( x = 2 \)**:
\[ p(2) = 2^3 - 6 \cdot 2^2 + 11 \cdot 2 - 6 \]
\[ = 8 - 24 + 22 - 6 = 0 \]
Hence, \( x = 2 \) is a zero of the polynomial.

3. **Check for \( x = 3 \)**:
\[ p(3) = 3^3 - 6 \cdot 3^2 + 11 \cdot 3 - 6 \]
\[ = 27 - 54 + 33 - 6 = 0 \]
Hence, \( x = 3 \) is a zero of the polynomial.

### Conclusion for Step 1:
Since \( p(1) = 0 \), \( p(2) = 0 \), and \( p(3) = 0 \), we can conclude that 1, 2, and 3 are indeed the zeroes of the polynomial \( p(x) \).

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### Step 2: Verify the Relation between Zeroes and Coefficients

Let the zeroes be \( \alpha = 1 \), \( \beta = 2 \), and \( \gamma = 3 \).

1. **Sum of the Zeroes**:
\[ \alpha + \beta + \gamma = 1 + 2 + 3 = 6 \]
According to Vieta's formulas, this should equal \( -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} = -\frac{-6}{1} = 6 \).

2. **Sum of the Product of the Zeroes Taken Two at a Time**:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = (1 \cdot 2) + (2 \cdot 3) + (3 \cdot 1) = 2 + 6 + 3 = 11 \]
According to Vieta's formulas, this should equal \( \frac{\text{coefficient of } x}{\text{coefficient of } x^3} = \frac{11}{1} = 11 \).

3. **Product of the Zeroes**:
\[ \alpha \beta \gamma = 1 \cdot 2 \cdot 3 = 6 \]
According to Vieta's formulas, this should equal \( -\frac{\text{constant term}}{\text{coefficient of } x^3} = -\frac{-6}{1} = 6 \).

### Conclusion for Step 2:
The relations hold true:
- Sum of zeroes \( = 6 \)
- Sum of products of zeroes taken two at a time \( = 11 \)
- Product of zeroes \( = 6 \)

Thus, we have verified both that 1, 2, and 3 are the zeroes of the polynomial and the relations between the zeroes and coefficients.

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