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 the powers of constant whole numbers (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 the power of a negative number,
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 the powers of constant whole numbers (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 the power of a negative number, 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 the powers of constant integers (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 study polynomials for a while, but most of this course will be 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.

Two algebraic expressions are **equivalent**
if they always lead to the same result when you evaluate them,
no matter what values you substitute for the variables.
For example, if *x* = 3,
then *x* + *x* + 4 = 3 + 3 + 4 = 10
and 2*x* + 4 = 2(3) + 4 = 10 also.
There's nothing special about 3 here;
the same thing would happen no matter what value we used for *x*,
so *x* + *x* + 4 is equivalent to 2*x* + 4.
(That's really what I meant
when I said above that they mean the same thing.)
In other words, *x* + *x* + 2 = 2*x* + 2
for every value of *x*.

When I say that you get the same result,
this includes the possibility that the result is undefined.
For example, 1/*x* + 1/*x* is equivalent to 2/*x*;
even when *x* = 0,
they both come out the same (in this case, undefined).
In other words, 1/*x* + 1/*x* = 2/*x*
whenever either side is defined,
which is the most that you could ask for.
In contrast, *x*^{2}/*x*
is *not* equivalent to *x*;
they *usually* come out the same,
but they are different when *x* = 0.
(Then *x*^{2}/*x* is undefined, but *x* is 0.)
So the best you can say in that case
is that *x*^{2}/*x* = *x*
whenever the left-hand side is defined,
or that *x*^{2}/*x* = *x*
for *x* ≠ 0.

Most of the identities that I mentioned before
are equivalences of algebraic expressions.
For example, *a* + *b* = *b* + *a*
for any real numbers *a* and *b*.
But to say that these are equal for *any* real numbers
is simply to say
that the expressions *a* + *b* and *b* + *a*
are equivalent.
On the other hand,
the identity that *a*/*a* = 1
for any nonzero real number *a*
is not quite an equivalence, since *a* cannot be zero.

So far, I've only used an identity
to evaluate each expression for the same value of the variables,
that is to substitute constants (specific numbers) for variables
in the identity.
So for example, if *a* = −3 and *b* = 4,
then I get −3 + 4 = 4 + (−3);
if you further remember that subtraction means adding the opposite,
this tells you how to calculate −3 + 4 as 4 − 3,
which is the number 1.
But in fact, identities are good for more than that,
and for Algebra we need to use them in more general ways.
In fact, given any equivalence of algebraic expressions,
you can get another equivalence
by substituting (not necessarily a constant but)
*any* defined algebraic expression for one of the variables
(and you can do this multiple times to substitute for multiple variables).
For example, if *a* = −3 (as before)
but now *b* = 2*x*,
then you get −3 + 2*x* = 2*x* − 3.
This is also an example
of the law that *a* + *b* = *b* + *a*,
just like −3 + 4 = 4 − 3 is,
but now it's an example that we'll need in Algebra.

A linear expression in **standard form**
is (essentially) the sum of one or more **terms**:
one **constant term**
and one term for each variable in the expression.
Furthermore, the term for a given variable
must be
the product of a constant **coefficient** and that variable,
while the constant term (of course) must be a constant.
Finally, the terms should come in alphabetical order (by the variable),
with the constant term last.
For example, in the linear expression 2*x* + 3*y* + 4,
the terms are 2*x*, 3*y*, and 4.
The first term, 2*x*, is the *x*-term, whose coefficient is 2.
The next term, 3*y*, is the *y*-term, whose coefficient is 3.
The constant term is 4;
we consider 4 to be its own coefficient,
on the grounds that 4 = 4 · 1.

There are some degenerate cases.
If the coefficient on any of the variables is 1, then we can omit it;
for example, 1*x* + 2 is equivalent to simply *x* + 2.
If any coefficient is negative,
then we use the opposite of a term with a positive coefficient,
which means that (unless this is the first term)
we write the sum using subtraction;
for example, 5*x* + (−3)*y* + 4
is equivalent to simply 5*x* − 3*y* + 4.
If the coefficient on any of the variables is −1,
then we combine the previous two rules;
for example, 5*x* + (−1)*y* + 4
is equivalent to simply 5*x* − *y* + 4.
Finally, if any coefficient is 0, then we can omit the entire term;
for example, 5*x* + 3*y* + 0
is equivalent to simply 5*x* + 3*y*.
(The exception to this is that if every coefficient is 0,
then we keep the constant term so that we have at least one term.)
These simplifications are considered part of the standard form.

A monomial in **standard form**
is (essentially) the product of one or more **factors**:
a constant **coefficient**
and one factor for each variable in the expression.
Furthermore, the factor for a given variable
must be
the variable raised to the power of a constant whole number,
the **degree** of that variable.
Finally, the factors should come in alphabetical order (by the variable),
with the coefficient (or **constant factor**) first.
For example, in the monomial 4*x*^{2}*y*^{3},
the factors are 4, *x*^{2}, and *y*^{3}.
First, the coefficient is 4.
The next factor, *x*^{2},
is the *x*-factor, whose degree is 2.
The last factor, *y*^{3},
is the *y*-factor, whose degree is 3.
We consider the degree of the coefficient (in this case, 4) to be 0
on the grounds that 4 = 4*x*^{0} for any *x*.

Again, there are some degenerate cases.
If the degree for any of the variables is 1, then we can omit it;
for example, 4*x*^{1}*y*^{3}
is equivalent to simply 4*x**y*^{3}.
Similarly, if the degree for any of the variables in 0,
then we can omit the entire factor;
for example, 4*x*^{0}*y*^{3}
is equivalent to simply 4*y*^{3}.
Alternatively, if the coefficient is 1, then we can omit it;
for example, 1*x*^{2}*y*^{3}
is equivalent to simply *x*^{2}*y*^{3}.
(The exception to this is that if every degree is 0
and the coefficient is 1,
then we keep the coefficient so that we have at least one factor.)
If the coefficient is negative,
then we use the opposite of the monomial with a positive coefficient;
for example, (−4)*x*^{2}*y*^{3}
is equivalent to simply −4*x*^{2}*y*^{3}.
If the coefficient is −1,
then we combine the previous two rules;
for example, (−1)*x*^{2}*y*^{3}
is equivalent to simply −*x*^{2}*y*^{3}.
Finally, if the coefficient is 0, then the entire monomial is zero;
for example, 0*x*^{2}*y*^{3}
is equivalent to simply 0.
Again, these simplifications are considered part of the standard form.

If the degree for a variable is 1, then we can leave the exponent off;
for example, 3*x*^{1} is equivalent to simply 3*x*.
If the degree for a variable is 0, then we can leave the variable out too;
for example, 2*x*^{0} is equivalent to simply 2.
If the coefficient is 1
(and there's at least one variable with a nonzero degree),
then we can leave it out;
for example, 1*x*^{3}
is equivalent to simply *x*^{3}.
Finally, if the coefficient is −1
(and there's at least one variable with a nonzero degree),
then we can replace it with a minus sign (indicating the additive inverse);
for example, −1*x*^{3}
is equivalent to simply −*x*^{3}.
These are also considered to be standard form;
in fact, you should always perform these simplifications,
at least when giving a final result.

The **degree** of a monomial
is the sum of the degrees for the variables in the monomial.
For example, the degree of 4*x*^{2}*y*^{3} is 5,
since 2 + 3 = 5.
Also, the degree of 4*x**y*^{3} is 4,
since the degree for *x* is 1 and 1 + 3 = 4.
Finally, the degree of a constant expression is 0,
since there is nothing to add;
remember that the degree of the coefficient itself is always zero.

Now, a polynomial in **standard form**
is the sum of one or more **terms**;
each term must be a monomial in standard form.
Also, the degrees of the terms should be decreasing as much as possible;
for terms with the same (total) degree,
the terms must have different degrees for at least one of the variables,
and the first variable (alphabetically) whose degree changes must decrease.
Finally, 0 should not be the coefficient of any term;
the exception to this is that if every coefficient is 0,
then we keep the constant term so that we have at least one term.

So a typical polynomial in standard form
is 2*x*^{3} + 4*x*^{2}*y* −
3*y*^{3} + *x*^{2} −
*x**y* + 1/2.
Notice that the respective degrees of these terms are 3, 3, 3, 2, 1, and 0;
among the degree-3 terms, the degrees on *x* are 3, 2, and 0.
The largest degree of the terms is 3;
we call this the **degree** of the polynomial as a whole.
Notice that the degree of a constatn expression is 0,
while the degree of a non-constant linear expression is 1.

It may not be obvious now, but every linear expression is equivalent to one in standard form, every monomial is equivalent to one in standard form, and every polynomial is equivalent to one in standard form. We'll see why next.

Remember that the terms of a polynomial are monomials.
Monomials are **like** (or *alike*)
if they are the same *except* (possibly) for their coefficients.
In other words, they're like
if they have the same variables with the same degrees.
For example, the monomials 3*x**y* and 5*x**y* are like;
they are both *x**y*-terms.
However, the monomial 2*x*^{2} is unlike these;
it is an *x*^{2}-term.
Evenm 2*x*^{2} and 2*x*^{3} are unlike,
because they have different degrees.

Remember that one of the fundamental identities for real numbers
says that (*a* + *b*)*c* =
*a**c* + *b**c*
for any real numbers *a*, *b*, and *c*.
If you substitute the coefficients of two terms for *a* and *b*
and substitute their common factor for *c*,
then this law (going backwards) tells you how to combine those terms.
For example,
if *a* = 3, *b* = −5, and *c* = *x*,
then I get (3 − 5)*x* = 3*x* − 5*x*;
since 3 − 5 = −2,
I can combine like terms
to get 3*x* − 5*x* = −2*x*.
In this way, you can put any polynomial into standard form
once it's given as the sum of monomials in standard form.
Just rearrange the terms in the correct order and combine the like terms.

Go back to the course homepage.

This web page was written in 2007 and 2008 by Toby Bartels. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-0950/2008w/standard/`

.