3X⁵-3X³+15X²

To factor the polynomial \(3X^5 - 3X^3 + 15X^2\), we can start by identifying the greatest common factor (GCF) of the coefficients and the variables.

1. The GCF of the coefficients \(3\), \(-3\), and \(15\) is \(3\).
2. The GCF of the variable parts \(X^5\), \(X^3\), and \(X^2\) is \(X^2\).

Thus, we can factor out \(3X^2\) from the polynomial.

Starting with the polynomial:

\[ 3X^5 - 3X^3 + 15X^2 \]

We factor out \(3X^2\):

\[ = 3X^2(X^3 - X + 5) \]

Now, we have factored the polynomial as:

\[ 3X^2(X^3 - X + 5) \]

The term \(X^3 - X + 5\) does not factor neatly over the reals (and finding roots might require the Rational Root Theorem or numerical methods).

Thus, the fully factored form is:

\[ 3X^2(X^3 - X + 5) \]