Polynomial

There are several conditions that need to be met in order to determine whether a mathematical expression is a polynomial. First, polynomials consist of constants and variables. These constants and variables make separable parts of a polynomial called terms and polynomials are made up of a finite number of these terms. The terms are separated by the symbols of mathematical operations (+,-,*), which brings us to the second condition.

The only mathematical operations which can be used in polynomials are addition, subtraction and multiplication. Division, in which the variable is a part of the denominator, does not produce a polynomial. If the denominator is a constant, a polynomial can be produced.

The third condition and the defining characteristic of polynomials are powers (or exponents). We covered exponents in the article called Squares and square roots, so if you are not sure what they are, feel free to refresh your memory. Each term in a polynomial has to have its non-negative integer exponent. That means that the value of the exponent for each term can be any integer between (and including) zero and any other finite value.

Terms and polynomials have their degrees or orders. The order of a term is determined by the sum of the exponents in that term. The order of a polynomial is the largest one among the orders of the terms that make the polynomial. That means that, for example, polynomials of the fourth order would look somewhat like this:

a*x4 + b*x3 + c*x2 + d*x + e

Or, if they have two variables, like this:

A*x2*y2 + b*x*y + c

We can name polynomials according to their degrees (or orders).

Degree

Name

0

constant

1

linear

2

quadratic

3

cubic

4

quartic (biquadratic)

5

quintic

6

sextic (hexic)

7

septic (heptic)

8

octic

9

nonic

10

decic

100

hectic

 

These are the degrees which have special names. For others you can always use the standard nomenclature – eleventh degree, 57th order and such.

Another important thing to know is that polynomials can have any number of terms as long as the number of terms is finite, no matter what their order is. Depending on that number, we name them:

Number of non-zero terms

Name

0

zero polynomial

1

monomial

2

binomial

3

trinomial

 

All other polynomials are simply called – polynomials.

What is interesting about the zero polynomial is that its order is intentionally left explicitly undefined or is defined to be negative, usually with the values of -1 or negative infinity. Also, it is the only one with an infinite number of roots.

Polynomials have several basic properties, a few of which we are going to mention here. The first one is that the sum of polynomials is always a polynomial (levitra online). Also the product of polynomials is itself a polynomial. The same thing goes for the composition of two polynomials.

Simplifying a polynomial

Solving polynomials of the third order or greater requires some advanced mathematical knowledge and we will show you how to do it when the time comes. But for now we will stick to the simplification of polynomials.

There are a few things you should have in mind when simplifying polynomials. The most important thing to remember is that addition and subtraction can be performed only with terms that have the same variables and of the same order! That is a general rule in mathematics, but it is particularly observable in this case. Also, all rules about the order of operations apply.

If after all this polynomials remind you of equations – you are right. In fact, polynomials perform as any other equations and you can treat them as such. But the reason we are mentioning them separately from equations is that they are the gateway to more complex mathematics and it is of vital importance that you understand them and know how to deal with them properly.

If you wish to practice what you learned and try to simplify a polynomial or two, please feel free to use the math worksheets below.

Polynomial exams for teachers

Exam Name File Size Downloads Upload date
Single variable – Integers
Integer polynomials with a single variable – very easy 462.4 kB 2510 October 13, 2012
Integer polynomials with a single variable – easy 462.6 kB 1332 October 13, 2012
Integer polynomials with a single variable – medium 462.5 kB 1779 October 13, 2012
Integer polynomials with a single variable – hard 465.2 kB 1597 October 13, 2012
Integer polynomials with a single variable – very hard 470.8 kB 1495 October 13, 2012
Single variable – Decimals
Decimal polynomials with a single variable – very easy 461.9 kB 963 October 13, 2012
Decimal polynomials with a single variable – easy 462.3 kB 857 October 13, 2012
Decimal polynomials with a single variable – medium 461.9 kB 868 October 13, 2012
Decimal polynomials with a single variable – hard 465.5 kB 785 October 13, 2012
Decimal polynomials with a single variable – very hard 468.7 kB 819 October 13, 2012
Single variable – Fractions
Polynomial fractions with a single variable – very easy 468.3 kB 1052 October 13, 2012
Polynomial fractions with a single variable – easy 466.9 kB 955 October 13, 2012
Polynomial fractions with a single variable – medium 468.1 kB 977 October 13, 2012
Polynomial fractions with a single variable – hard 473.7 kB 994 October 13, 2012
Polynomial fractions with a single variable – very hard 481.4 kB 995 October 13, 2012
Two variables – Integers
Integer polynomials with two variables – very easy 464.7 kB 1235 October 13, 2012
Integer polynomials with two variables – easy 464.6 kB 1046 October 13, 2012
Integer polynomials with two variables – medium 465 kB 1216 October 13, 2012
Integer polynomials with two variables – hard 469.3 kB 1203 October 13, 2012
Integer polynomials with two variables – very hard 474.1 kB 1102 October 13, 2012
Two variables – Decimals
Decimal polynomials with two variables – very easy 465 kB 817 October 13, 2012
Decimal polynomials with two variables – easy 465.6 kB 789 October 13, 2012
Decimal polynomials with two variables – medium 465 kB 793 October 13, 2012
Decimal polynomials with two variables – hard 470.5 kB 800 October 13, 2012
Decimal polynomials with two variables – very hard 475.9 kB 899 October 13, 2012
Two variables – Fractions
Polynomial fractions with two variables – very easy 470.3 kB 844 October 13, 2012
Polynomial fractions with two variables – easy 470.4 kB 757 October 13, 2012
Polynomial fractions with two variables – medium 471.1 kB 782 October 13, 2012
Polynomial fractions with two variables – hard 479.3 kB 767 October 13, 2012
Polynomial fractions with two variables – very hard 487.3 kB 902 October 13, 2012


Polynomial worksheets for students

Worksheet Name File Size Downloads Upload date
Single variable – Integers
Simplify 4 integer polynomials with a single variable 911.5 kB 2044 October 14, 2012
Simplify 5 integer polynomials with a single variable 996.2 kB 972 October 14, 2012
Simplify 6 integer polynomials with a single variable 1 MB 1065 October 14, 2012
Simplify 7 integer polynomials with a single variable 1.2 MB 976 October 14, 2012
Simplify 8 integer polynomials with a single variable 1.2 MB 912 October 14, 2012
Simplify 9 integer polynomials with a single variable 1.3 MB 1660 October 14, 2012
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 927.6 kB 1053 October 14, 2012
Simplify 5 decimal polynomials with a single variable 1 MB 1149 October 14, 2012
Simplify 6 decimal polynomials with a single variable 747.7 kB 817 October 14, 2012
Simplify 7 decimal polynomials with a single variable 1.2 MB 1020 October 14, 2012
Simplify 8 decimal polynomials with a single variable 1.3 MB 756 October 14, 2012
Simplify 9 decimal polynomials with a single variable 1.3 MB 888 October 14, 2012
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 4.2 MB 993 October 14, 2012
Simplify 5 polynomial fractions with a single variable 11.7 MB 853 October 14, 2012
Simplify 6 polynomial fractions with a single variable 1.3 MB 916 October 14, 2012
Simplify 7 polynomial fractions with a single variable 2.7 MB 766 October 14, 2012
Simplify 8 polynomial fractions with a single variable 1.9 MB 1085 October 14, 2012
Simplify 8 polynomial fractions with a single variable 2.1 MB 819 October 14, 2012
Two variables – Integers
Simplify 4 integer polynomials with two variables 1 MB 2092 October 14, 2012
Simplify 5 integer polynomials with two variables 1.2 MB 1221 October 14, 2012
Simplify 6 integer polynomials with two variables 1.3 MB 1082 October 14, 2012
Simplify 7 integer polynomials with two variables 1.4 MB 973 October 14, 2012
Simplify 8 integer polynomials with two variables 1.5 MB 1174 October 14, 2012
Simplify 9 integer polynomials with two variables 1.6 MB 1173 October 14, 2012
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 1.1 MB 841 October 14, 2012
Simplify 5 decimal polynomials with two variables 1.2 MB 917 October 14, 2012
Simplify 6 decimal polynomials with two variables 1.4 MB 968 October 14, 2012
Simplify 7 decimal polynomials with two variables 1.4 MB 743 October 14, 2012
Simplify 8 decimal polynomials with two variables 1.5 MB 769 October 14, 2012
Simplify 9 decimal polynomials with two variables 1.6 MB 792 October 14, 2012
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 10.1 MB 1104 October 14, 2012
Simplify 5 polynomial fractions with two variables 4.2 MB 994 October 14, 2012
Simplify 6 polynomial fractions with two variables 1.3 MB 755 March 5, 2012
Simplify 7 polynomial fractions with two variables 2 MB 777 October 14, 2012
Simplify 8 polynomial fractions with two variables 2.2 MB 976 October 14, 2012
Simplify 9 polynomial fractions with two variables 2.3 MB 1058 October 14, 2012