Chapter 8
Special Factorizations
Multiply two binomials and the work is routine. Do it enough times and something shows up: a handful of products keep landing in the same shape, whatever numbers you started with. Factoring, meanwhile, has always meant hunting, trying pairs until one fits. A shape you have seen a hundred times should not need hunting. It should be recognizable on sight, read backward from the product to the factors it came from. A short list of these shapes is worth knowing by heart, and the surprising part is how far a few of them reach.
What You'll Explore
5 lessons.
- Squares of Binomials
Squaring a single number is quick work. Squaring a sum of two terms is a product you will write out again and again, and after a few times the answer starts to look predictable. This lesson asks what that pattern actually is, and whether it can be read in reverse to recognize where a trinomial came from.
- Difference of Squares
Most expressions with only two terms refuse to come apart at all. One family is the exception, and it factors every single time, always along the same lines. This lesson takes up why that family is special, whether a sum gets the same courtesy, and how the pattern turns an awkward product into mental arithmetic.
- Sum and Difference of Cubes
Cubes might be expected to behave like squares, only with a higher power. Whether that expectation holds is the first question here. This lesson looks at which sums and differences of perfect cubes come apart, what the pieces look like when they do, and how to keep track of the signs in an answer with more parts than before.
- Rationalizing Denominators
Two fractions can have exactly the same value while one carries a square root on the bottom and the other does not. Which one counts as simplified, and why should it matter? You will look at the convention behind that choice, and at what can be done to a denominator without changing what the fraction is worth.
- Factoring by Grouping
Plenty of polynomials fit none of the patterns in this chapter, and an expression with four terms offers no obvious place to begin. Here you will see whether a polynomial can be persuaded to show a common factor it did not appear to have, and what that has to do with the trinomials that resisted guessing.