Factoring Rules

Here are a few basic factoring examples which occur often enough to justify memorizing them. Again, recognition will be the key to using these rules in Calculus.

1. a - b = ( a - b)( a + b)

2. ( a + b) = a + 2 ab + b

3. ( a - b) = a - 2 ab + b

4. ( a + b) = a + 3 a b + 3 ab + b

5. ( a - b) = a - 3 a b + 3 ab - b

6. a - b = ( a - b)( a + ab + b)

As you might expect, the last three are used less often than the first three, since they involve third powers. In fact, number 1 is probably the most important, since we often run into the difference of two perfect squares, especially in geometric problems.

Completing the Square

Given x + bx or x + bx + c, it is sometimes useful to rewrite it in the form ( x + a) + d using rule 2 (or 3 if b is negative), for some constants a and d related to b (and c). This is called completing the square, and it’s where the quadratic formula comes from.

Let so b = 2 a. Then x + bx = x + 2 ax which looks like two of the three pieces of rule 2. The last piece of rule 2 would be adding a, but to keep things fair we must also subtract a 2 so we don’t change anything. This gives us

x + bx = x + 2 ax = ( x + 2 ax + a) - a = ( x + a) - a so and

If we started with x + bx + c, then the c just comes along for the ride and we get

x + bx + c = x + 2ax + c = (x + 2ax + a) - a + c = (x + a) + (c - a)

so didn't change, but now .

Examples:

x + 6x + 7 = (x + 2(3)x + 9) + (7 - 9) = (x + 3) - 2