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:
- Factoring Common Factor
- Factoring by Grouping and Regrouping
- Factoring Quadratics
- Difference of Squares
- Sum and Difference of Cubes
To be successful in factoring polynomials, you need to recognize when and what method to use.
Example 1. Factor `2x^3-8x` completely.
`=2x(x^2-4)=` (factor out `2x`)
`=2x(x-2)(x+2)` (apply difference of squares formula)
You possibly need to perform more than two steps.
Example 2. Factor completely: `-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)
Let's solve one more example.
Example 3. Factor `x^12-1` completely.
`=(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.)
Now, it is time to exercise.
Exercise 1. Factor `8x^8+125x^2` completely.
Exercise 2. Factor completely: `-2y^3-16y^2-30y`.
Exercise 3. Factor `512x^9-y^9` completely.