My Favorite Identities (In Context)

Back when I taught college algebra, I liked to tell students what I see when I look at a complicated equation. Perhaps it will help others.

The first thing I look for are the major "chunks" or terms that are separated by a plus sign, a minus sign, or an equals sign. I would write things like this on the board


3(x-1)^2/(45x) - (10x+4)/2 + 55 = 1/(x^2-1)


and tell them to circles the "chunks" or terms. I told them to always include the sign with the term. If a term doesn't have a sign, then the sign is '+'.


Identity [a = 1 * a]

25 = (+1) * 25

If you don't see a sign, it's a plus. Always remember that 1 is a factor.


The first term is

+3(x-1)^2/(45x)


The second term is

-(10x+4)/2


The third term is

+55


and the fourth term is

+1/(x^2-1)


This is the first level of breakdown of any equation. You have to see the terms.


You can move any term (no matter how complicated) from one side of the equals sign to the other by changing its sign. Remember to change and take the sign of the term with you. None of the other terms change at all when you do this.

I haven't said what I wanted to do with this equation. I'm just seeing what I can do if I want.


All terms are fractions (rational expressions) and have a numerator and a denominator.

For the first term, the numerator is +3(x-1)^2 and the denominator is (45x). The third term has a numerator of 55 and a denominator of 1. If you don't explicitly see a denominator, it is 1.


Identity [a = a/1]

This is the second level of breakdown for any equation. You have to see the fractions.


Next, look at each term (numerator and denominator) and find all of the factors. These are things that are multiplied together. Remember that exponents belong to the factor.
For the first term, the numerator has (3) and (x-1)^2 as factors. (I suppose that you could say that (x-1)^2 is really two factors, (x-1) and (x-1), but I don't think that kind of factoring is necessary. Keeping the exponent has its uses.) The denominator has the factors (45) and (x). If you see something like x-5 all by itself in a numerator or denominator, I really, really like to see it surrounded by parens. This makes it clear that it is a factor and must be treated as a whole. Students do weird things with something like "x-5" if you don't enclose it with parens.


This is the third level of breakdown for any equation. You have to see the factors.


You can change the position of any factors in the numerator or any factors in the denominator.

Identity [a*b = b*a]

Term 1 could be:

+(x-1)^2*3/(x*45)


Which brings up another issue. Mathematicians like to order factors in a particular way. This is not required, but it's a good thing to know. The simplest rule is to put the number factors (constants) first and the variables after that. Some like to sort the factors based on the size of the exponents, or they like to keep the variables in the same order for all terms.


All factors (actually, all numbers and variables) have exponents.

Identity [a = a^1] (This is not really part of the basic math identity set)

If you don't see an exponent, it's a one.


You can move any factor from the denominator to the numerator, or vice versa, by changing the sign of the exponent.


Identity [x^(-a) = 1/x^a]

1/(x-2) = (x-2)^(-1)


(x-2) is a factor. Its exponent is 1. I can move it to the numerator by changing the sign of the exponent. You need to be able to make these changes (for whatever reason) without a second thought. It has to be automatic.


With these basic identities and breakdown, you can push terms and factors around to meet your needs. This is the lowest level of equation manipulation.