An **algebraic expression**
is any grammatically sensible expression
made up of any or all of the following:

- specific numbers (called
**constants**); - letters (or other symbols) standing for numbers
(called
**variables**); and - arithmetic operations.

In this course, we'll mostly be dealing with linear expressions.
A **linear expression**
is an algebraic expression made up *only* from any or all of these:

- Constants;
- Variables;
- Addition;
- Multiplication by constants;
- Taking opposites (optional);
- Subtraction (optional);
- Division by nonzero constants (optional).

Typical linear expressions
are things like
*x* + 2, 2*t* + 4, and 5*x* − *y*.
But notice some things that you *can't* do in a linear expression:
you can't divide by a non-constant expression,
or use exponentiation, or take absolute values.
So for example, 1/*x*, *y*^{2}, and |*a*|
are *not* linear expressions.

A **monomial**
is an algebraic expression made up *only* from any or all of these:

- Constants;
- Variables;
- Multiplication;
- Taking opposites (optional);
- Division by nonzero constants (optional);
- Raising to constant whole exponents (optional).

Typical monomials
are things like
*x*^{2}, 2*x**y*,
and −*x**y*^{3}/5.
But again some things you *can't* do in a monomial:
you can't add or subtract, or divide by a non-constant expression,
or raise to a negative exponent, and you still can't take absolute values.
So for example, *x* + 5, 1/*x*,
*y*^{−2}, and |*a*|
are *not* monomials.

A **polynomial**
is an algebraic expression made up *only* from any or all of these:

- Constants;
- Variables;
- Addition;
- Multiplication;
- Taking opposites (optional);
- Subtraction (optional);
- Division by nonzero constants (optional);
- Raising to constant whole exponents (optional).

Typical polynomials
include the linear expressions and monomials above,
as well as things like
*x*^{2} + 5
and 2*x**y*^{2} + 3*x*.
But there are still some things you *can't* do in a polynomial:
you still can't divide by a non-constant expression,
raise to a negative exponent, or take absolute values.
So for example, 1/*x*, *y*^{−2}, and |*a*|
are still *not* polynomials.

Finally, a **rational expression**
is an algebraic expression made up *only* from any or all of these:

- Constants;
- Variables;
- Addition;
- Multiplication;
- Taking reciprocals;
- Taking opposites (optional);
- Subtraction (optional);
- Division (optional);
- Raising to constant integers as exponents (optional).

Typical rational expressions include all the examples above,
as well as things like
1/*x*, *y*^{−2},
and even
(2*x*^{2} + 5)/(4*x**y* − 7*z*).
But there are still a few things
that you *can't* do in a rational expression;
most of these are operations that you won't study until Intermediate Algebra,
but you *still* can't take absolute values.
So for example, |*a*| is still *not* a rational expression.

We really won't use rational expression in this course, except as occasional examples. We'll use polynomials a bit, but mostly this course is about linear expressions. However, we'll also study expressions that are almost linear but include absolute values; I don't know a special name for those.

The instruction to substitute 3 for *x*
is often written as ‘*x* = 3’,
because it's based on the idea
that the variable *x* is equal to the number 3.
But a more precise notation is ‘*x* := 3’,
which indicates not only that *x* is equal to 3
but also that you want to *use* that fact in a certain way.
The book uses the simpler notation (without the colon)
and it's all right with me if you do that too,
but I'm going to use the colon.

Notice that sometimes evaluating an expression
shows it to be undefined.
For example, if you have 5/*t* and you substitute *t* := 0,
then you get 5/0, which is undefined.
That's OK; just say that it's undefined, that's the answer.

An expression can have more than one variable in it;
to evaluate the expression,
you must substitute something for *each* variable.
For example, in the expression *x* + *y*,
if you substitute *x* := 3, you get 3 + *y*,
which is still not evaluated.
But if you substitute *x* := 3 *and* *y* := 4,
then you get 3 + 4, which evaluates to 7.
You can indicate both these substitutions at once
as (*x* := 3, *y* := 4)
or as (*x*, *y*) := (3, 4).

Different variables
can be substituted with the same value or with different values,
but a single variable
can only be substituted with *one* value in any given situation.
For example, in the expression *x* + *x*^{2},
you can substitute *x* := 3
and get 3 + 3^{2} (which evaluates to 12),
or you can substitute *x* := 4
and get 4 + 4^{2} (which evaluates to 20),
but it makes no sense to use 3 + 4^{2}
(where the first *x* is 3 and the second is 4).
The value of *x* may vary from situation to situation,
but it can only be one thing at a time!

Often you want to put in extra parentheses when you substitute.
For example, if you have 2 · *x*
and you substitute *x* := 4,
then you get 2 · 4, which is fine (and works out to 8).
But if you abbreviate 2 · *x* as 2*x*
(as we usually do in Algebra)
and again substitute *x* := 4,
you *don't* want to say simply 24!
Instead, you must either put the dot back and use 2 · 4 again,
or else put in some parentheses and use 2(4), which means the same thing.
As another example, in the expression *a*^{2},
if you substitute *a* := 3,
then you get 3^{2}, which is fine (and works out to 9).
But if you substitute *a* := −3 instead,
then you *don't* want to say −3^{2},
because this means the opposite of 3^{2},
which is *not* want you want!
Instead, you should say (−3)^{2},
which means −3 times −3, which is what you want.
That is, the correct answer is 9, *not* −9.
If in doubt,
use parentheses (or another grouping symbol) when substituting.

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