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.

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-7-6/”>Answer Key 7.6