A **polynomial**
is an algebraic expression built out of constants and variables
using the operations of addition and multiplication,
or anything equivalent to this.
Thus, anything built using addition, subtraction, opposites, multiplication,
division by nonzero constants, and raising to powers of whole constants
is 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.)
However, it is forbidden in a polynomial
to divide by zero or a variable expression
(or to take the reciprocal of zero or a variable expression),
to raise to the power of a negative or fractional number or of a variable,
or to take an absolute value.

Usually the first step in dealing with any polynomial
is to simplify it to **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).

Every polynomial is a sum of **terms**;
you add up the terms to get the entire expression.
Here are a couple of examples:

- 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.

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
is a monomial with 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.

- 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 even −∞, meaning negative infinity, but I will say that the degree is undefined.)

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.
For example, neither 2*x*^{2} + 3*x* + 4
nor *x*^{3} − 2*x*^{2} + 3,
but 2*x* + 3*y* + 4 is linear,
as is *a* − 2*b* + 3.
Most of the first half of this course will be about linear expressions.

A polynomial is **quadratic** if its degree is 2,
or **cubic** if its degree is 3.
Thus, 2*x*^{2} + 3*x* + 4 is quadratic,
while *x*^{3} −
2*x*^{2} + 3 is cubic.
There are additional words (now based on Latin words for the numbers)
for polynomials of degree 4, 5, or more,
but these are not used much.
(Even ‘cubic’ is not used in the textbook.)
There doesn't seem to be a word for a polynomial whose degree is 1
other than ‘nonconstant linear’,
and there doesn't seem to be a word for a polynomial whose degree is 0
other than ‘nonzero constant’.

Notice that a monomial is simply a polynomial with 1 term.
Similarly, a **binomial** is a polynomial with 2 terms,
a **trinomial** is a polynomial with 3 terms,
and a **tetranomial** is a polynomial with 4 terms.
For example,
both 2*x*^{2} + 3*x* + 4
and *x*^{3} − 2*x*^{2} + 3
are trinomials.
There are additional words (based on Greek words for the numbers)
for polynomials with 5, 6, or more terms,
but these are not used much.
(Even ‘tetranomial’ is not used in the textbook.)

- 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*.

Combining like terms defines **addition** of polynomials;
for example:

- 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.

- The opposite of 2
*x*^{2}+ 3*x*+ 4 is −2*x*^{2}− 3*x*− 4; - The opposite
of
*x*^{3}− 2*x*^{2}+ 3 is −*x*^{3}+ 2*x*^{2}− 3.

- To subtract 2
*x*^{2}+ 3*x*+ 4 from*x*^{3}− 2*x*^{2}+ 3, we add −2*x*^{2}− 3*x*− 4 to*x*^{3}− 2*x*^{2}+ 3 to get*x*^{3}− 4*x*^{2}− 3*x*− 1.

- 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}.

Combining like factors defines **multiplication** of monomials;
for example:

- 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

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This web page was written between 2007 and 2018 by Toby Bartels, last edited on 2018 July 26. Toby reserves no legal rights to it.

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