Doing algebra amounts to playing with symbols. Some people actually think it’s fun. Other people tend to be a little intimidated by it. We’ll see that the knowledge of some basic identities can help us out in doing this algebra. Here’s a simple motivational example that I’ll use to show you the importance of knowing some identities early on. It is asked here to sketch the graph of the function phi defined by the formula x squared minus 2x minus 1. Now, certainly we could plot a bunch of points and sketch a graph of the function. But there’s a smarter way to do it that uses a little bit of algebra and it uses the following observation.
If we look at the expression x minus 1 and square it, we get x squared minus 2x plus 1. This equality is the key to writing the formula for phi in a more useful way. Using it, we can write phi of x in the form x minus 1 squared minus 2. So we have managed to write phi in a different way. What’s the advantage of having done that? Well we see that the function phi will be closely related, its graph will be closely related, to that of the function x squared, which, of course, we already know. It’s our famous parabola. The x has been replaced by x minus 1, which means there is a translation in the argument.
As we’ve seen, that means the graph of the function is shifted horizontally 1 unit. Furthermore, the minus 2 corresponds to a translation of the values of the function. That will correspond to a shift downwards in the graph of the function. And now we have the graph of the function phi without doing any work but by doing a little bit of algebra. Some notable identities that every mathematician knows are the following. Now, I should say that an identity is not just an equality that holds for certain numbers. The word identity means it holds for all possible values of here, a and b. And this is actually the identity I used a moment ago to help me rewrite the function phi.
Because, if in the general identity you replace a by x and b by minus 1, then you get exactly the equality that I invoked before. Another identity worth knowing is for the difference of squares. A squared minus b squared can be written as the product a plus b times a minus b. There is a notable identity for the difference of cubes as well and for the sum of two cubes. Now let’s look at polynomials. They are arguably the most important class of functions in mathematics. The function phi that we were looking at a moment ago is an example of a polynomial. The general definition is the following.
A polynomial is one whose defining formula can be expressed as the sum of a certain number of terms. There is a term constant an times x to the power n, that’s the highest power that will occur, and then a term with the next lower power, all the way down to a1 times x, and finally just a number a0 at the end. This is called a polynomial of degree n, n being the highest power that occurs. Of course, for that power to actually be there, the an should be different from 0. Otherwise it would vanish. The real numbers ai, a0 up to an, are called the coefficients of the polynomial. An is called the leading coefficient.
And a0 is called the constant term. So, for example, the function phi that we have seen is a polynomial of degree 2. Its leading term is x squared. And, therefore, the leading coefficient is 1. And the constant term is minus 1. An affine function, that is one of the form mx plus b, is a polynomial of degree 1 provided that the slope m is different from 0. If m is equal 0, then it’s a constant function, which we think of as being a polynomial of degree 0. You notice that a polynomial has terms in it that are defined for any real number x. So the natural domain of a polynomial is the whole real line. Now, we can combine polynomials.
We can add them. We can multiply by a constant. We can subtract. Here’s just a simple example to give the idea quickly. If we want to simplify this expression, we see that it is actually a certain polynomial p1 initially plus twice another polynomial minus 3 times a third. How do we simplify? Well, we remove the parentheses and then we group all the terms as a function of what the power is. So there’s going to be an x cubed term, a single one. There’s going to be a term with x squared to which two other terms will contribute. Similarly, there’s going to be a term in x and there’s going to be, finally, the constant term, the constants together.
When you do all of that and simplify, you get the canonical form of the polynomial which is, in this case, x cubed minus 2x minus 12. Now, you may have remarked that in this example and in others the coefficients of our polynomials were integers. That’s often true in applications as well. But it doesn’t have to be true. In general, the coefficients are just real numbers. We can also multiply polynomials. Here’s an example. We want to multiply these two polynomials together. How do we do it? We just use the usual distributive law, except that we’re doing it with symbols rather than actual numbers. We then group all the powers of x together and we come down to the canonical answer.
A rational function means a function that can be expressed as the quotient of two polynomials, that is p over q. Here’s an example of how it might occur. We have here two rational functions that we wish to add and express as a single one. How do you add these two fractions? Well, the key is to do exactly the same thing that you would do if you were dealing with numbers instead of symbols. You need a lowest common denominator. In this case, you want the denominator of each part of the addition to have the factor x minus 2 and the factor x squared plus 1. That can easily be arranged.
Once you have the same denominator, then you simply add the two fractions by adding their numerators. And, of course, some simplification may be necessary and you arrive at the final answer. Later on we’ll even show how to divide one polynomial by another. You’ll see how much fun that is.