Polynomials
Algebraic Expressions: The Building Blocks of Algebra
An algebraic expression is a collection of numerals, letters, grouping
symbols, and operators (such as +, , Â·, and
Ã·).
Here are some examples of algebraic expressions:
The algebraic expression 5x^{2}  12  xy^{3} has three terms:
5x^{2}, 12, and xy^{3}
A term in an algebraic expression is a quantity joined to other quantities
by the operation of addition or subtraction.
The term 12 is called a constant term because it does not contain a
variable. Note that when a term is subtracted, a negative sign is attached to
the individual term.
The numeric factor of a term is called the coefficient of the variables of
that term.
For example, in the algebraic expression 5x^{2}  12  xy^{3}, the coefficient of
x^{2} is 5 and the coefficient of xy^{3} is 1.
Note:
In the algebraic expression x^{2}, the coefficient
of x^{2} is 1.
x^{2} = 1x^{2}
Polynomials
A polynomial is a special kind of algebraic expression.
In a polynomial, each variable must have a whole number exponent.
Here are some examples of polynomials.
8x + 5 
2x^{3}  wy^{2} + 7x  6 
24w^{2}y 
25 
Note:
The polynomial 8x + 5
can be written 8x^{1} + 5x^{0}.
An algebraic expression is not a polynomial if a variable is in the
denominator, inside absolute value bars, or under a radical sign.
For example, these algebraic expressions are not polynomials:
Polynomials with one, two, or three terms have special names.
Name 
Number of terms 
Examples 


monomial binomial
trinomial 
1 2
3 
x,
x + 7,
x^{2}  5x + 7, 
9wy^{2}, 6x^{2}  5,
5x^{3}y^{2} + 6xy  9 
13 4w^{2}y
10wy^{3} 
The degree of a term of a polynomial is the sum of the exponents of the
variables in that term. For example,
â€¢ the degree of 8x^{2}y^{4} is 6 because 2 + 4 = 6;
â€¢ the degree of 4^{3}y^{2} is 2. Note that 4 is not a variable so its
exponent, 3, does not affect the degree of the term.
Note:
In the expression 4^{3}y^{2}, the exponent 3 does
not contribute to the degree of the term.
That's because 3 is the exponent of a
constant, not a variable.
The degree of a polynomial is equal to the degree of the term with the
highest degree.
For example, the degree of 8x^{2}y^{4}  5x^{3}y + 4^{3}y^{2} is 6 because the term with
the highest degree has degree 6.
Note:
In 8x^{2}y^{4}  5x^{3}y + 4^{3}y^{2}, the term with the
highest degree is 8x^{2}y^{4}. It has degree 2 + 4 = 6.
Example
Find the degree of this polynomial: 13w^{4}y^{3} + 5wy^{4}
 y^{2} + 32
Solution
First, find the degree of each term.
The term with the highest degree is 13w^{4}y^{3}. The degree of this term is 7.
Therefore, the degree of the polynomial is 7.
