- 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.

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 (no variables).
Then there is one factor for each variable,
which consists of that variable raised to the power of a whole number.
This whole number
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 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.

- 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

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