An **variable**
is an expression that stands for an unknown or unspecified number;
we usually use a letter from the English alphabet,
such as *a*, *x*, or *V*.
(Uppercase and lowercase letters count as different variables.)
An **algebraic expression**
is an expression built (in a meaningful way)
out of constants, variables, and arithmetic operations,
such as *x* + 5, *x**y*^{2}, and 8/*p*.
A variable by itself also counts as an algebraic expression,
as does an arithmetic expression with no variables.
To **evaluate** an algebraic expression,
you need to be given values of each of the variables in the expression;
for example, if you evaluate *x* + 5 when *x* is 7,
then the result is 7 + 5 = 12,
but if you evaluate the same expression when *x* is 9,
then the result is 9 + 5 = 14.

Two expressions are **equivalent**
if they always evaluate to the same result
whenever you use the same values for all of the variables;
for example, *x* + 5 and 5 + *x* are equivalent.
To **simplify** an algebraic expression,
you find an equivalent expression in some standard form;
for example, *x* + 5 is considered standard
while 5 + *x* is not,
so 5 + *x* simplifies to *x* + 5.
Exactly what counts as the standard form of an expression
will depend on the context that you're working in,
so ‘simplify’ doesn't always mean to make things simpler
(but it usually at least doesn't make them any more complicated).

There are many different types of algebraic expressions,
but most of the expressions in this class are a particular type: a polynomial.
A **polynomial**
is an algebraic expression built out of constants and variables
using the operations of addition and multiplication,
or anything equivalent to this;
anything built using addition, subtraction, opposites, multiplication,
division by nonzero constants, and raising to powers of whole constants
is a polynomial,
but it is forbidden
to divide by zero or a variable expression,
to raise to the power of negative constant or a variable,
or to take an absolute value
is forbidden in a polynomial.
(This is because subtraction is adding an opposite,
you can take an opposite by multiplying by −1,
dividing by a nonzero constant is multiplying by its reciprocal,
raising to the power of zero is equivalent to the constant 1,
and raising to the power of a nonzero whole number
is equivalent to repeated multiplication.)

Usually the first step in dealing with any polynomial is to simplify it to standard form.

- The terms of 2
*x*^{2}+ 3*x*+ 4 are 2*x*^{2}, 3*x*, and 4; - The terms
of
*x*^{3}− 2*x*^{2}+ 3 are*x*^{3}, −2*x*^{2}, and 3.

Each term of a polynomial is a monomial;
a **monomial**
is any expression built out of constants and variables
using *only* the operation of multiplication,
or anything equivalent to this.
Anything built using
opposites, multiplication, division by nonzero constants,
and raising to powers of whole constants
is a monomial,
but addition or subtraction is forbidden.

Every monomial is a product of **factors**;
you multiply the factors to get the entire term.
The first factor is the **coefficient**,
which is always a constant.
Then there is one factor for each variable,
which consists of that variable raised to the power of a whole constant.
This whole constant
is the **degree** of the monomial on that variable.
So in the end, a monomial is given by several numbers:
the coefficient and the degrees on all of the variables.

Think about these special cases before we look at some examples:

- If there is no constant factor, then the coefficient is 1.
- If there is no exponent on a variable, then the degree on that variable is 1.
- If a variable does not appear, then the degree on that variable is 0.

Now here are some examples:

- The coefficient of 2
*x*^{2}is 2, and its degree on*x*is 2. - The coefficient of 3
*x*is 3, and its degree on*x*is 1. (Remember that*x*^{1}=*x*, so we can think of 3*x*as 3*x*^{1}.) - The coefficient of 4 is 4, and its degree on
*x*is 0. (Remember that*x*^{0}= 1, multiplication by which does nothing, so we can think of 4 as 4*x*^{0}.) - The coefficient of
*x*^{3}is 1, and its degree on*x*is 3. (Remember that multiplication by 1 does nothing, so we can think of*x*^{3}as 1*x*^{3}.) - The coefficient of −2
*x*^{2}is −2, and its degree on*x*is 2. - The coefficient of 3 is 3, and its degree on
*x*is 0.

Now if you look back at the first two examples of polynomials,
we see that each of these polynomials has three terms,
and each of these terms has a coefficient and a degree on *x*.

So far, I've only talked about degrees on specific variables.
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 the degree on*x*is 2 and the degree on*y*is 3 (and 2 + 3 = 5). - The degree of 4
*x**y*^{3}is 4, since the degree on*x*is 1 (and 1 + 3 = 4). - The degree of the constant monomial 7 is 0, since there is nothing to add; the degree on every variable is 0.

When writing out a polynomial,
we always write the terms in order of decreasing degree.
The **degree** of the polynomial as a whole
is the largest degree of any of its nonzero terms.
Returning to our original examples:

- The degrees
of the terms of 2
*x*^{2}+ 3*x*+ 4 are 2, 1, and 0, so the degree of this polynomial is 2. - The degree
of
*x*^{3}− 2*x*^{2}+ 3 is 3, since the degrees of its terms are 3, 2, and 0.

A polynomial whose degree is zero is equivalent to a constant.
(Since the constant 0 has no nonzero terms,
it technically does not have a degree at all;
sometimes people say that its degree is −1 or −∞.)
A **linear expression**
is a polynomial whose degree is at most 1;
in other words, either it is constant or its degree is exactly 1.
A linear expression can be built out of constants and variables
using only addition and multiplication by constants;
any expression built from addition, subtraction, opposites,
multiplication by constants, and division by constants
is linear,
but multiplication by a variable expression
or raising to a power (other than 0 or 1)
is forbidden.

- In 2
*x*+ 3*x*, 2*x*and 3*x*are alike, since they are both*x*-terms; - In 2
*x*+ 2*y*, 2*x*and 2*y*are*not*alike, since one is an*x*-term but the other is a*y*-term; - In 2
*x*+ 2*x*^{2}, 2*x*and 2*x*^{2}are also*not*alike, since one is an*x*-term but the other is an*x*^{2}-term.

- 2
*x*+ 3*x*= 5*x*.

- To add 2
*x*^{2}+ 3*x*+ 4 and*x*^{3}− 2*x*^{2}+ 3, we get*x*^{3}+ 0*x*^{2}+ 3*x*+ 7, which is*x*^{3}+ 3*x*+ 7.

Similarly, two factors are **alike**
(or **like factors**)
if they are both coefficients or involve the same variable.
For example:

- In
*x*^{2}*x*^{3},*x*^{2}and*x*^{3}are alike, since they are both*x*-factors; - In
*x*^{2}*y*^{2},*x*^{2}and*y*^{2}are*not*alike, since one is an*x*-factor but the other is a*y*-factor; - In 2
*x*^{2}, 2 and*x*^{2}are also*not*alike, since one is a coefficient but the other is an*x*-factor.

*x*^{2}*x*^{3}=*x*^{5}.

- To multiply 2
*x*^{2}*y*^{3}and*x*^{3}*y*^{2}, we get 2*x*^{5}*y*^{5}.

- To multiply 2
*x*^{2}+ 3*x*+ 4 and*x*^{3}− 2*x*^{2}+ 3, we get:- (2
*x*^{2})(*x*^{3}) + (2*x*^{2})(−2*x*^{2}) + (2*x*^{2})(3) + (3*x*)(*x*^{3}) + (3*x*)(−2*x*^{2}) + (3*x*)(3) + (4)(*x*^{3}) + (4)(−2*x*^{2}) + (4)(3), which is - 2
*x*^{5}− 4*x*^{4}+ 6*x*^{2}+ 3*x*^{4}− 6*x*^{3}+ 9*x*+ 4*x*^{3}− 8*x*^{2}+ 12, which is - 2
*x*^{5}− 4*x*^{4}+ 3*x*^{4}− 6*x*^{3}+ 4*x*^{3}+ 6*x*^{2}− 8*x*^{2}+ 9*x*+ 12, which is - 2
*x*^{5}−*x*^{4}− 2*x*^{3}− 2*x*^{2}+ 9*x*+ 12.

- (2

For now, we're only going to add polynomials and multiply them by constants; this includes taking their opposites (which is multiplying by the constant −1) and subtracting them (which is adding their opposites). We'll return to multiplying more generally (and also some special cases of division) at the end of the course.

Go back to the course homepage.

This web page was written between 2007 and 2016 by Toby Bartels, last edited on 2016 July 28. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-0950/2016SU/polynomials/`

.