Finding the Zeroes of the Polynomial \(p(x) = x^2 - 3x\)

The zeroes of a polynomial are the values of the variable (\(x\) in this case) for which the polynomial evaluates to zero. To find the zeroes of the polynomial \(p(x) = x^2 - 3x\), we set \(p(x)\) equal to 0 and solve for \(x\).

Given polynomial:

\(p(x) = x^2 - 3x\)

Set \(p(x) = 0\) to find the zeroes:

\(x^2 - 3x = 0\)

We can factor out the common term, which is \(x\), from both terms on the left side of the equation:

\(x(x - 3) = 0\)

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have two possible cases:

1. The first factor, \(x\), is equal to zero.
2. The second factor, \((x - 3)\), is equal to zero.

Case 1:

\(x = 0\)

Case 2:

\(x - 3 = 0\)

To solve for \(x\) in the second case, add 3 to both sides of the equation:

\(x = 3\)

So, the values of \(x\) for which the polynomial \(p(x) = x^2 - 3x\) is equal to zero are \(x = 0\) and \(x = 3\).

Thus, the zeroes of the polynomial \(p(x) = x^2 - 3x\) are 0 and 3.

Comparing this with the given options, the correct option is the one listing 0 and 3.