# Using Techniques for Factoring Together

## Related Calculator: Factoring Polynomials Calculator

Now, it is time to understand how to apply learned techniques together.

Recall, that we've learned following factoring techniques:

To be successful in factoring polynomials, you need to recognize when and what method to use.

Example 1. Factor 2x^3-8x completely.

2x^3-8x=

=2x(x^2-4)= (factor out 2x)

=2x(x-2)(x+2) (apply difference of squares formula)

Answer: 2x^3-8x=2x(x-2)(x+2).

You possibly need to perform more than two steps.

Example 2. Factor completely: -y^4-y^2+2.

-y^4-y^2+2=

=-(y^4+y^2-2)= (factor out -1)

=-(y^2+2)(y^2-1)= (factor quadratics)

=-(y^2+2)(y-1)(y+1)= (apply difference of squares formula)

Answer: -y^4-y^2+2=-(y^2+2)(y-1)(y+1).

Let's solve one more example.

Example 3. Factor x^12-1 completely.

x^12-1=(x^4)^3-1^3=

=(x^4-1)(x^8+x^4+1)= (difference of cubes)

=(x^2-1)(x^2+1)(x^8+x^4+1)= (difference of squares)

=(x-1)(x+1)(x^2+1)(x^8+x^4+1) (difference of squares once more.)

Answer: x^12-1=(x-1)(x+1)(x^2+1)(x^8+x^4+1).

Now, it is time to exercise.

Exercise 1. Factor 8x^8+125x^2 completely.

Answer: x^2(2x^2+5)(4x^4-10x^2+25).

Exercise 2. Factor completely: -2y^3-16y^2-30y.

Answer: -2y(y+3)(y+5).

Exercise 3. Factor 512x^9-y^9 completely.

Answer: (2x-y)(4x^2+2xy+y^2)(64x^6+8x^3y^3+y^6).