Determine the zeroes of the polynomial `p(x)=x^(3)-2x^(2)`. Also verify the relationship between the zeroes and the coefficient.

Answer

To determine the zeroes of the polynomial \( p(x) = x^3 - 2x^2 \) and verify the relationship between the zeroes and the coefficients, we can follow these steps:

### Step 1: Set the polynomial equal to zero
We start by setting the polynomial equal to zero to find the zeroes:
\[ p(x) = x^3 - 2x^2 = 0 \]

### Step 2: Factor the polynomial
We can factor out the common term \( x^2 \):
\[ x^2(x - 2) = 0 \]

### Step 3: Solve for zeroes
Now, we can set each factor equal to zero:
1. \( x^2 = 0 \) gives \( x = 0 \) (with multiplicity 2)
2. \( x - 2 = 0 \) gives \( x = 2 \)

Thus, the zeroes of the polynomial are:
\[ x = 0, 0, 2 \]

### Step 4: Verify the relationship between the zeroes and the coefficients
The polynomial can be expressed in the standard form:
\[ p(x) = ax^3 + bx^2 + cx + d \]
Here, \( a = 1 \), \( b = -2 \), \( c = 0 \), and \( d = 0 \).

#### Sum of the zeroes
The sum of the zeroes \( \alpha + \beta + \gamma \) is given by:
\[ \alpha + \beta + \gamma = -\frac{b}{a} = -\frac{-2}{1} = 2 \]
Calculating the left-hand side:
\[ 0 + 0 + 2 = 2 \]
Thus, the sum of the zeroes matches the coefficient relationship.

#### Sum of the products of the zeroes taken two at a time
The sum of the products of the zeroes taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha \) is given by:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{0}{1} = 0 \]
Calculating the left-hand side:
\[ (0 \cdot 0) + (0 \cdot 2) + (2 \cdot 0) = 0 \]
Thus, this relationship also holds.

#### Product of the zeroes
The product of the zeroes \( \alpha\beta\gamma \) is given by:
\[ \alpha\beta\gamma = -\frac{d}{a} = -\frac{0}{1} = 0 \]
Calculating the left-hand side:
\[ 0 \cdot 0 \cdot 2 = 0 \]
Thus, this relationship is verified as well.

### Conclusion
All three relationships between the zeroes and the coefficients have been verified:
1. Sum of zeroes: \( 0 + 0 + 2 = 2 \)
2. Sum of products of zeroes taken two at a time: \( 0 = 0 \)
3. Product of zeroes: \( 0 = 0 \)