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)`.