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When factoring polynomials, the first step is often to identify the greatest common factor (GCF). The GCF is the largest factor that divides each term of the polynomial without leaving a remainder. In our example, we have the polynomial \(10x^2 - 10y^2\).
Both terms \(10x^2\) and \(10y^2\) share a common factor of 10.
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This means we can factor out 10 from each term. By factoring out the GCF, you simplify the polynomial to 10 \((x^2 - y^2)\).

Factoring out the GCF is a powerful technique because it breaks down the polynomial into a simpler form, making the next steps easier. Always look for the GCF first; it can often save a lot of time and effort in more complex problems.

After factoring out the GCF, you will often encounter special forms such as the difference of squares. A difference of squares is a polynomial in the form \(a^2 - b^2\). It can be factored into \((a + b)(a - b)\).

In our example, once we factored out the 10, we have \(x^2 - y^2\), which is a perfect example of a difference of squares.

Recognizing this form allows you to apply the formula directly.

Here, \(a = x\) and \(b = y\). So, we can factor \(x^2 - y^2\) into \((x + y)(x - y)\).
This simplifies our previous expression to 10 \((x + y)(x - y)\).

The difference of squares is a handy pattern to recognize since it often appears in polynomial factoring problems.

Not every polynomial can be factored into simpler expressions. When a polynomial cannot be factored further using integers, it's called a prime polynomial.

Identifying prime polynomials early on can save you time. If a polynomial is already in a simple form and does not fit any common factoring patterns (like GCF or difference of squares), it might be prime. In our exercise, once we have factored the polynomial into 10 \((x + y)(x - y)\), each factor \((x + y)\) and \((x - y)\) cannot be factored further using integers, and hence are already in their simplest form.

Knowing when to stop factoring is just as important as knowing how to factor. Always check for the GCF and special forms first, and if no further factoring is possible, you have likely reached the prime form of the polynomial.