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Factoring polynomials is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, give you back the original polynomial. This is an essential technique for solving polynomial equations.
We start by looking for the greatest common factor (GCF), which is the largest factor that each term in the polynomial shares. In our original exercise, the function was:
$$f(x)=x^{3}-2x^{2}-3x$$
Here, the GCF is 'x', since each term in the polynomial has an 'x' as a factor. Dividing each term by 'x' gives us:
$$x(x^2 - 2x - 3)$$Next, we factor the quadratic polynomial inside the parentheses. This means finding two numbers that multiply to give the constant term (-3) and add up to give the coefficient of the middle term (-2). These numbers are -3 and 1, so we rewrite and factor the quadratic as:
$$x(x - 3)(x + 1)$$Factoring is not only useful for solving equations but also for simplifying expressions and finding points where the polynomial is equal to zero.

The greatest common factor (GCF) is the highest factor that can divide each term within a polynomial without a remainder. Identifying the GCF is typically the first step in factoring a polynomial.
In the given exercise, the original polynomial is:
$$x^{3}-2x^{2}-3x$$
To find the GCF, we look at all the terms and notice that each term has at least one 'x' in it. So, the GCF is 'x'. By factoring out 'x' from each term, we simplify the polynomial to:
$$x(x^2 - 2x - 3)$$
The importance of the GCF cannot be overstated. Removing the GCF simplifies the polynomial and makes the next steps of solving much easier.
Always make sure you have correctly identified and factored out the GCF before attempting any further factoring.

Solving polynomial equations involves finding all values of 'x' that make the equation equal to zero. These values are also called the 'zeros' or 'roots' of the polynomial.
After factoring the polynomial in our exercise, we got:
$$x(x - 3)(x + 1) = 0$$
To find the zeros, we set each factor equal to zero and solve for 'x':
$$x = 0$$
$$x - 3 = 0 \rightarrow x = 3$$
$$x + 1 = 0 \rightarrow x = -1$$
Thus, the solutions or zeros of the polynomial equation are:
$$x = 0, x = 3, x = -1$$
This means the function 'f(x)' crosses the x-axis at these points.
Understanding how to solve polynomial equations by factoring and setting each factor to zero is a crucial skill in algebra.