Polynomials

The parts of an expression connected by the signs + or -- are called
the terms of the expression. For example, $x^{2}$, $2xy$,
and $\sqrt{x}$ are the terms of the expression $x^{2}+2xy+\sqrt{x}$,
and the expression $2a-ab+c^{2}$ is made up of three terms: $2a$,
$-ab$, and $c^{2}$.

A polynomial or more precisely a polynomial in $x$
is an algebraic expression consisting of terms in the form $ax^{k}$
where $k$ is a nonnegative integers (that is zero or a natural number
0, 1, 2, ... ) and $a$ is a real number called the coefficient
of the term. For example,

$ 3,\quad x,\quad7x^{51},\quad-x^{3}+\frac{1}{\sqrt{14}}x-3,\quad\text{and }\quad7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3 $

are all polynomials. In the last example, the coefficients are $7,-\pi,-\sqrt{2},-1,1,$
and $-3$. Note that any constant is also a polynomial because it
can be written as $ax^{0}$; for example $3=3x^{0}$.

• Of course, instead of a polynomial in $x$, we may have polynomials
  in $y$ or $z$ or any other letter. For example, $4z^{3}-0.5z+1$
  is a polynomial in $z$.
• A polynomial that has only one term is called a monomial.
  For example, $4x^{3}$ and $-\sqrt{7}x^{2}$ are two monomials. A
  polynomial that has two terms is called a binomial, and a
  polynomial that has three terms is called a trinomial.
• The largest exponent in a polynomial is called the degree
  of the polynomial. In $7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3$,
  the largest power of $x$ and hence the degree of the polynomial is

Polynomials of degree 0, 1, 2, and 3 have special names. If $a\neq0$
then

name	form	degree\tabularnewline constant polynomial	$a$	0\tabularnewline linear polynomial	$ax+b$	1\tabularnewline quadratic polynomial	$ax^{2}+bx+c$	2\tabularnewline cubic polynomial	$ax^{3}+bx^{2}+cx+d$	3\tabularnewline

• When two or more terms differ only in their numerical coefficients,
  we say they are similar or like terms. For example,
  $4x^{2}$, $\frac{3}{2}x^{2}$, and $-\sqrt{3}x^{2}$ are like terms
  but $3x$ and $3x^{2}$ are unlike terms.