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 0 B 3015 January 1, 1970
Integer polynomials with a single variable – easy 0 B 1611 January 1, 1970
Integer polynomials with a single variable – medium 0 B 2162 January 1, 1970
Integer polynomials with a single variable – hard 0 B 1852 January 1, 1970
Integer polynomials with a single variable – very hard 0 B 1751 January 1, 1970
Single variable – Decimals
Decimal polynomials with a single variable – very easy 0 B 1119 January 1, 1970
Decimal polynomials with a single variable – easy 0 B 1015 January 1, 1970
Decimal polynomials with a single variable – medium 0 B 1015 January 1, 1970
Decimal polynomials with a single variable – hard 0 B 898 January 1, 1970
Decimal polynomials with a single variable – very hard 0 B 952 January 1, 1970
Single variable – Fractions
Polynomial fractions with a single variable – very easy 0 B 1230 January 1, 1970
Polynomial fractions with a single variable – easy 0 B 1094 January 1, 1970
Polynomial fractions with a single variable – medium 0 B 1134 January 1, 1970
Polynomial fractions with a single variable – hard 0 B 1127 January 1, 1970
Polynomial fractions with a single variable – very hard 0 B 1248 January 1, 1970
Two variables – Integers
Integer polynomials with two variables – very easy 0 B 1452 January 1, 1970
Integer polynomials with two variables – easy 0 B 1244 January 1, 1970
Integer polynomials with two variables – medium 0 B 1428 January 1, 1970
Integer polynomials with two variables – hard 0 B 1383 January 1, 1970
Integer polynomials with two variables – very hard 0 B 1318 January 1, 1970
Two variables – Decimals
Decimal polynomials with two variables – very easy 0 B 977 January 1, 1970
Decimal polynomials with two variables – easy 0 B 923 January 1, 1970
Decimal polynomials with two variables – medium 0 B 936 January 1, 1970
Decimal polynomials with two variables – hard 0 B 909 January 1, 1970
Decimal polynomials with two variables – very hard 0 B 1022 January 1, 1970
Two variables – Fractions
Polynomial fractions with two variables – very easy 0 B 999 January 1, 1970
Polynomial fractions with two variables – easy 0 B 880 January 1, 1970
Polynomial fractions with two variables – medium 0 B 927 January 1, 1970
Polynomial fractions with two variables – hard 0 B 895 January 1, 1970
Polynomial fractions with two variables – very hard 0 B 1058 January 1, 1970

Polynomial worksheets for students

Worksheet Name File Size Downloads Upload date
Single variable – Integers
Simplify 4 integer polynomials with a single variable 0 B 2436 January 1, 1970
Simplify 5 integer polynomials with a single variable 0 B 1101 January 1, 1970
Simplify 6 integer polynomials with a single variable 0 B 1228 January 1, 1970
Simplify 7 integer polynomials with a single variable 0 B 1093 January 1, 1970
Simplify 8 integer polynomials with a single variable 0 B 1037 January 1, 1970
Simplify 9 integer polynomials with a single variable 0 B 1858 January 1, 1970
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 0 B 1166 January 1, 1970
Simplify 5 decimal polynomials with a single variable 0 B 1249 January 1, 1970
Simplify 6 decimal polynomials with a single variable 0 B 916 January 1, 1970
Simplify 7 decimal polynomials with a single variable 0 B 1122 January 1, 1970
Simplify 8 decimal polynomials with a single variable 0 B 854 January 1, 1970
Simplify 9 decimal polynomials with a single variable 0 B 984 January 1, 1970
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 0 B 1150 January 1, 1970
Simplify 5 polynomial fractions with a single variable 0 B 959 January 1, 1970
Simplify 6 polynomial fractions with a single variable 0 B 1023 January 1, 1970
Simplify 7 polynomial fractions with a single variable 0 B 887 January 1, 1970
Simplify 8 polynomial fractions with a single variable 0 B 1255 January 1, 1970
Simplify 8 polynomial fractions with a single variable 0 B 936 January 1, 1970
Two variables – Integers
Simplify 4 integer polynomials with two variables 0 B 2933 January 1, 1970
Simplify 5 integer polynomials with two variables 0 B 1419 January 1, 1970
Simplify 6 integer polynomials with two variables 0 B 1304 January 1, 1970
Simplify 7 integer polynomials with two variables 0 B 1203 January 1, 1970
Simplify 8 integer polynomials with two variables 0 B 1342 January 1, 1970
Simplify 9 integer polynomials with two variables 0 B 1356 January 1, 1970
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 0 B 931 January 1, 1970
Simplify 5 decimal polynomials with two variables 0 B 1001 January 1, 1970
Simplify 6 decimal polynomials with two variables 0 B 1067 January 1, 1970
Simplify 7 decimal polynomials with two variables 0 B 879 January 1, 1970
Simplify 8 decimal polynomials with two variables 0 B 869 January 1, 1970
Simplify 9 decimal polynomials with two variables 0 B 883 January 1, 1970
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 0 B 1265 January 1, 1970
Simplify 5 polynomial fractions with two variables 0 B 1088 January 1, 1970
Simplify 6 polynomial fractions with two variables 0 B 866 January 1, 1970
Simplify 7 polynomial fractions with two variables 0 B 887 January 1, 1970
Simplify 8 polynomial fractions with two variables 0 B 1067 January 1, 1970
Simplify 9 polynomial fractions with two variables 0 B 1180 January 1, 1970