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 3554 September 3, 2019
Integer polynomials with a single variable – easy 462.6 kB 1945 September 3, 2019
Integer polynomials with a single variable – medium 462.5 kB 2584 September 3, 2019
Integer polynomials with a single variable – hard 465.2 kB 2202 September 3, 2019
Integer polynomials with a single variable – very hard 470.8 kB 2108 September 3, 2019
Single variable – Decimals
Decimal polynomials with a single variable – very easy 461.9 kB 1311 September 3, 2019
Decimal polynomials with a single variable – easy 462.3 kB 1204 September 3, 2019
Decimal polynomials with a single variable – medium 461.9 kB 1207 September 3, 2019
Decimal polynomials with a single variable – hard 465.5 kB 1088 September 3, 2019
Decimal polynomials with a single variable – very hard 468.7 kB 1151 September 3, 2019
Single variable – Fractions
Polynomial fractions with a single variable – very easy 468.3 kB 1452 September 3, 2019
Polynomial fractions with a single variable – easy 466.9 kB 1286 September 3, 2019
Polynomial fractions with a single variable – medium 468.1 kB 1360 September 3, 2019
Polynomial fractions with a single variable – hard 473.7 kB 1300 September 3, 2019
Polynomial fractions with a single variable – very hard 481.4 kB 1490 September 3, 2019
Two variables – Integers
Integer polynomials with two variables – very easy 464.7 kB 1730 September 3, 2019
Integer polynomials with two variables – easy 464.6 kB 1489 September 3, 2019
Integer polynomials with two variables – medium 465 kB 1700 September 3, 2019
Integer polynomials with two variables – hard 469.3 kB 1644 September 3, 2019
Integer polynomials with two variables – very hard 474.1 kB 1583 September 3, 2019
Two variables – Decimals
Decimal polynomials with two variables – very easy 465 kB 1166 September 3, 2019
Decimal polynomials with two variables – easy 465.6 kB 1117 September 3, 2019
Decimal polynomials with two variables – medium 465 kB 1103 September 3, 2019
Decimal polynomials with two variables – hard 470.5 kB 1089 September 3, 2019
Decimal polynomials with two variables – very hard 475.9 kB 1254 September 3, 2019
Two variables – Fractions
Polynomial fractions with two variables – very easy 470.3 kB 1179 September 3, 2019
Polynomial fractions with two variables – easy 470.4 kB 1042 September 3, 2019
Polynomial fractions with two variables – medium 471.1 kB 1105 September 3, 2019
Polynomial fractions with two variables – hard 479.3 kB 1065 September 3, 2019
Polynomial fractions with two variables – very hard 487.3 kB 1245 September 3, 2019

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 2867 September 3, 2019
Simplify 5 integer polynomials with a single variable 996.2 kB 1340 September 3, 2019
Simplify 6 integer polynomials with a single variable 1 MB 1455 September 3, 2019
Simplify 7 integer polynomials with a single variable 1.2 MB 1295 September 3, 2019
Simplify 8 integer polynomials with a single variable 1.2 MB 1280 September 3, 2019
Simplify 9 integer polynomials with a single variable 1.3 MB 2115 September 3, 2019
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 927.6 kB 1411 September 3, 2019
Simplify 5 decimal polynomials with a single variable 1 MB 1423 September 3, 2019
Simplify 6 decimal polynomials with a single variable 747.7 kB 1102 September 3, 2019
Simplify 7 decimal polynomials with a single variable 1.2 MB 1293 September 3, 2019
Simplify 8 decimal polynomials with a single variable 1.3 MB 1028 September 3, 2019
Simplify 9 decimal polynomials with a single variable 1.3 MB 1173 September 3, 2019
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 4.2 MB 1386 September 3, 2019
Simplify 5 polynomial fractions with a single variable 11.7 MB 1131 September 3, 2019
Simplify 6 polynomial fractions with a single variable 1.3 MB 1200 September 3, 2019
Simplify 7 polynomial fractions with a single variable 2.7 MB 1040 September 3, 2019
Simplify 8 polynomial fractions with a single variable 1.9 MB 1438 September 3, 2019
Simplify 8 polynomial fractions with a single variable 2.1 MB 1101 September 3, 2019
Two variables – Integers
Simplify 4 integer polynomials with two variables 1 MB 3647 September 3, 2019
Simplify 5 integer polynomials with two variables 1.2 MB 1662 September 3, 2019
Simplify 6 integer polynomials with two variables 1.3 MB 1541 September 3, 2019
Simplify 7 integer polynomials with two variables 1.4 MB 1391 September 3, 2019
Simplify 8 integer polynomials with two variables 1.5 MB 1581 September 3, 2019
Simplify 9 integer polynomials with two variables 1.6 MB 1645 September 3, 2019
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 1.1 MB 1080 September 3, 2019
Simplify 5 decimal polynomials with two variables 1.2 MB 1135 September 3, 2019
Simplify 6 decimal polynomials with two variables 1.4 MB 1217 September 3, 2019
Simplify 7 decimal polynomials with two variables 1.4 MB 1023 September 3, 2019
Simplify 8 decimal polynomials with two variables 1.5 MB 1004 September 3, 2019
Simplify 9 decimal polynomials with two variables 1.6 MB 1026 September 3, 2019
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 10.1 MB 1464 September 3, 2019
Simplify 5 polynomial fractions with two variables 4.2 MB 1226 September 3, 2019
Simplify 6 polynomial fractions with two variables 1.3 MB 1047 September 3, 2019
Simplify 7 polynomial fractions with two variables 2 MB 1063 September 3, 2019
Simplify 8 polynomial fractions with two variables 2.2 MB 1213 September 3, 2019
Simplify 9 polynomial fractions with two variables 2.3 MB 1425 September 3, 2019