Polynomials
A term is an expression containing a number or the product of a number
and one or more variables raised to powers. Some examples of terms are
4x^{3}, x^{2}y^{3}, 6ab, and 2.
A polynomial is a single term or a finite sum of terms. The powers of
the variables in a polynomial must be positive integers. For example,
4x^{3} + (15x^{2}) + x + (2)
is a polynomial. Because it is simpler to write addition of a negative as
subtraction, this polynomial is usually written as
4x^{3} 15x^{2} + x 2
The degree of a polynomial in one variable is the highest power of the
variable in the polynomial. So 4x^{3} 15x^{2} + x 2 has degree
3 and 7w w^{2} has degree 2.
The degree of a term is the power of the variable in the term. Because
the last term has no variable, its degree is 0.
4x^{3} 
15x^{2} 
+ x 
2 
↓ 
↓ 
↓ 
↓ 
Thirddegree term 
Seconddegree term 
Firstdegree term 
Zerodegree term 
A single number is called a constant and so the last term is the constant
term. The degree of a polynomial consisting of a single number such as 8 is 0.
The number preceding the variable in each term is called the coefficient of a
variable or the coefficient of that term. In 4x^{3} 15x^{2} + x
2 the coefficient of x^{3} is 4, the coefficient of x^{2} is
15, and the coefficient of x is 1 because x = 1 Â· x.
Identifying coefficients
Determine the coefficients of x^{3} and x^{2} in each
polynomial:
a) x^{3} + 5x^{2}  6
b) 4x^{6}  x^{3} + x
Solution
a) Write the polynomial as 1 Â· x^{3} + 5x^{2}  6 to see that
the coefficient of x^{3} is 1 and the coefficient of x^{2} is 5.
b) The x^{2}term is missing in 4x^{6}  x^{3} + x.
Because 4x^{6}  x^{3} + x can be written as
4x^{6}  1 Â· x^{3} + 0 Â· 4x^{6}  x^{2 }+ x,
the coefficient of x^{3 }is 1 and the coefficient of x^{2
}is 0.
For simplicity we generally write polynomials with the exponents decreasing
from left to right and the constant term last. So we write
x^{3}  4x^{2} + 5x + 1 rather than  4x^{2} +
1 + 5x + x^{3}
When a polynomial is written with decreasing exponents, the coefficient of
the first term is called leading coefficient.
Certain polynomials are given special names. A monomial is a
polynomial that has one term, a binomial is a polynomial that has two
terms, and a trinomial is a polynomial that has three terms. For example,
3x^{5} is a monomial, 2x  1 is a binomial, and 4x^{6} 3x + 2
is a trinomial.
Types of polynomials
Identify each polynomial as a monomial, binomial, or trinomial and state its
degree.
a) 5x^{2}  7x^{3} + 2
b) x^{43}  x^{2}
c) 5x
d) 12
Solution
a) The polynomial 5x^{2}  7x^{3} + 2 is a thirddegree
trinomial.
b) The polynomial x^{43}  x^{2} is a binomial with degree
43.
c) Because 5x = 5x^{1}, this polynomial is a monomial with degree 1.
c) The polynomial 12 is a monomial with degree 0.
