Find the zeroes of the polynomial `p(x)=x(x-2) (x+3)`

Answer

To find the zeroes of the polynomial \( p(x) = x(x - 2)(x + 3) \), we will follow these steps:

### Step 1: Set the Polynomial Equal to Zero
To find the zeroes of the polynomial, we need to set \( p(x) \) equal to zero:
\[ p(x) = 0 \]
This gives us:
\[ x(x - 2)(x + 3) = 0 \]

### Step 2: Apply the Zero Product Property
According to the zero product property, if the product of several factors is zero, at least one of the factors must be zero. Therefore, we can set each factor equal to zero:
1. \( x = 0 \)
2. \( x - 2 = 0 \)
3. \( x + 3 = 0 \)

### Step 3: Solve Each Equation
Now, we will solve each equation for \( x \):

1. From \( x = 0 \):
\[ x = 0 \]

2. From \( x - 2 = 0 \):
\[ x = 2 \]

3. From \( x + 3 = 0 \):
\[ x = -3 \]

### Step 4: List the Zeroes
The zeroes of the polynomial \( p(x) = x(x - 2)(x + 3) \) are:
\[ x = 0, \quad x = 2, \quad x = -3 \]

### Final Answer
Thus, the zeroes of the polynomial are \( 0, 2, \) and \( -3 \).

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