Skill: Factor polynomial expressions completely

To factor a polynomial completely, rewrite it as a product of factors. Use this order:

1. Find the greatest common factor (GCF).
2. Look at what remains for a familiar pattern, such as a difference of squares.
3. Check that none of the factors can be factored further.

Example

Factor completely:

$$6x^3-24x$$

Step 1: Find the GCF.

The GCF of $6x^3$ and $24x$ is $6x$. Factor it out:

$$6x^3-24x=6x(x^2-4)$$

You can check this by distributing:

$$6x(x^2)-6x(4)=6x^3-24x$$

Step 2: Factor the expression inside the parentheses.

The expression $x^2-4$ is a difference of squares because

$$x^2-4=x^2-2^2$$

Use the pattern

$$a^2-b^2=(a-b)(a+b)$$

So,

$$x^2-4=(x-2)(x+2)$$

Step 3: Write the complete factorization.

$$\boxed{6x(x-2)(x+2)}$$

This is completely factored because $6$ is a constant, $x$ is a single variable factor, and neither $x-2$ nor $x+2$ can be factored further using integer factors.