Chapter 7: Factoring

# 7.6 Factoring Quadratics of Increasing Difficulty

Factoring equations that are more difficult involves factoring equations and then checking the answers to see if they can be factored again.

Example 7.6.1

Factor .

This is a standard difference of squares that can be rewritten as , which factors to . This is not completely factored yet, since can be factored once more to give .

Therefore, .

This multiple factoring of an equation is also common in mixing differences of squares with differences of cubes.

Example 7.6.2

Factor .This is a standard difference of squares that can be rewritten as , which factors to . This is not completely factored yet, since both and can be factored again. and This means that the complete factorization for this is: Example 7.6.3

A more challenging equation to factor looks like . This is not an equation that can be put in the factorable form of a difference of squares. However, it can be put in the form of a sum of cubes. In this form, factors to .

Therefore, .

Example 7.6.4

Consider encountering a sum and difference of squares question. These can be factored as follows: factors as a standard difference of squares as shown below: Simplifying inside the brackets yields: Which reduces to: Therefore: Examples 7.6.5

Consider encountering the following difference of cubes question. This can be factored as follows: factors as a standard difference of squares as shown below:  Simplifying inside the brackets yields: Sorting and combining all similar terms yields: Therefore, the result is: # Questions

Completely factor the following equations.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.  