# Working with algebraic expressions: Introduction

What is the coefficient of $x^2$ in the polynomial $x^3 + 17 x^2 +3x + 2$?

## Working with algebraic expressions: Introduction

When you deal with algebraic expressions, it's easy to get confused by all those letters and numbers floating around. It helps a lot if you keep your expressions tidy and as simple as possible. Here's an algebraic expression with three terms. Four X, Plus two times X, Minus X. The first thing you can do when you see something like this, is to get rid of that multiplication sign, between two and X.

The number that stands in front of a variable tells you how many of that variable you have. We call this number the coefficient of the variable. There's no need for a multiplication sign between the coefficient and the variable. In this example, all the three terms have the same variable -- they are all of the same kind. That means we can go ahead and add them together.

Four X plus two X is six X. But then what? The third X-term doesn't have a coefficient -- it's just an X! If there's no coefficient spelled out, it means there's one of that variable. X means one X.

So we subtract that last X, and get a total of five X. And there you go! We have simplified the expression to five X. Here's another algebraic expression. This one has four terms: four Y, Plus five, Minus two Y, Plus two.

Begin by bringing together terms that are of the same kind. First we gather all the Y:s. Four Y minus two Y equals two Y. Then we sum up the constants separately. The constants are the numbers that don't have any variables at all.

Five plus two equals seven. We say that we have combined like terms. That means we have brought together all terms that are of the same kind; apples with apples and pears with pears. The algebraic expression we have left has two different types of term, one Y-term and one constant. Let's do another one.

Two M, Plus M squared Minus five. Are there any like terms here? No there isn't! M, and M squared, are not the same kind of term, so we can't add them together. Just like you can't add together the distance two meters with the area one square meter, you can't add M and M squared.

They are different units, different kinds of term. So in this expression, all the like terms are already gathered. But there is one thing we can do. It helps to sort the terms, putting the one with the largest exponent first. So we rearrange it like this.

First the M-squared term. Then the M term. And then the constant at the end. This doesn't change the value of the expression at all. But if you always sort expressions like this, you make them easier to read and interpret.

An expression like this, with several types of term, is called a polynomial. Poly means many: Many types of term. And this particular polynomial is of the second degree, because it has one variable to the second power. It's a second degree polynomial. Take a good look at it, because you'll meet plenty more of these.