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Second Example

Let's try another example which requires factoring in steps 1 and 2:

5x3 - 10x2 - 15x

Again, the three steps in Factoring Completely are:

  1. Factor a GCF from the expression, if possible.
  2. Factor a Trinomial, if possible.
  3. Factor a Difference Between Two Squares as many times as possible.

Step 1

We see that the terms in our example have a greatest common factor of 5x. As instructed, we will factor out this GCF:

5x(x2 - 2x - 3)

Step 2

We see that (x2 - 2x - 3) is a factorable trinomial, so we factor it:

5x(x + 1)(x - 3)

Proceeding to Step 3, we can look over our expression and see that neither 5x, nor (x + 1), nor (x - 3) can be factored as a difference between two squares. We have factored 5x3 - 10x2 - 15x completely.

Final Example

For our final example, we will make use of all three Factoring Completely steps.

12x4 - 3x2 - 54

Step 1

We factor out a Greatest Common Factor of 3.

3(4x4 - x2 - 18)

Step 2

3(4x2 - 9)(x2 + 2)

Step 3

Finally, we identify (4x2 - 9) as a binomial that can be factored into (2x + 3)(2x - 3). So the completely factored result is

3(2x + 3)(2x - 3)(x2 + 2).

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