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. 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 1846 October 13, 2012
Integer polynomials with a single variable – easy 462.6 kB 990 October 13, 2012
Integer polynomials with a single variable – medium 462.5 kB 1274 October 13, 2012
Integer polynomials with a single variable – hard 465.2 kB 1206 October 13, 2012
Integer polynomials with a single variable – very hard 470.8 kB 1148 October 13, 2012
Single variable – Decimals
Decimal polynomials with a single variable – very easy 461.9 kB 704 October 13, 2012
Decimal polynomials with a single variable – easy 462.3 kB 662 October 13, 2012
Decimal polynomials with a single variable – medium 461.9 kB 691 October 13, 2012
Decimal polynomials with a single variable – hard 465.5 kB 640 October 13, 2012
Decimal polynomials with a single variable – very hard 468.7 kB 652 October 13, 2012
Single variable – Fractions
Polynomial fractions with a single variable – very easy 468.3 kB 808 October 13, 2012
Polynomial fractions with a single variable – easy 466.9 kB 715 October 13, 2012
Polynomial fractions with a single variable – medium 468.1 kB 744 October 13, 2012
Polynomial fractions with a single variable – hard 473.7 kB 756 October 13, 2012
Polynomial fractions with a single variable – very hard 481.4 kB 749 October 13, 2012
Two variables – Integers
Integer polynomials with two variables – very easy 464.7 kB 903 October 13, 2012
Integer polynomials with two variables – easy 464.6 kB 807 October 13, 2012
Integer polynomials with two variables – medium 465 kB 941 October 13, 2012
Integer polynomials with two variables – hard 469.3 kB 937 October 13, 2012
Integer polynomials with two variables – very hard 474.1 kB 841 October 13, 2012
Two variables – Decimals
Decimal polynomials with two variables – very easy 465 kB 584 October 13, 2012
Decimal polynomials with two variables – easy 465.6 kB 640 October 13, 2012
Decimal polynomials with two variables – medium 465 kB 652 October 13, 2012
Decimal polynomials with two variables – hard 470.5 kB 634 October 13, 2012
Decimal polynomials with two variables – very hard 475.9 kB 737 October 13, 2012
Two variables – Fractions
Polynomial fractions with two variables – very easy 470.3 kB 646 October 13, 2012
Polynomial fractions with two variables – easy 470.4 kB 607 October 13, 2012
Polynomial fractions with two variables – medium 471.1 kB 644 October 13, 2012
Polynomial fractions with two variables – hard 479.3 kB 618 October 13, 2012
Polynomial fractions with two variables – very hard 487.3 kB 678 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 1472 October 14, 2012
Simplify 5 integer polynomials with a single variable 996.2 kB 784 October 14, 2012
Simplify 6 integer polynomials with a single variable 1 MB 847 October 14, 2012
Simplify 7 integer polynomials with a single variable 1.2 MB 810 October 14, 2012
Simplify 8 integer polynomials with a single variable 1.2 MB 745 October 14, 2012
Simplify 9 integer polynomials with a single variable 1.3 MB 1295 October 14, 2012
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 927.6 kB 830 October 14, 2012
Simplify 5 decimal polynomials with a single variable 1 MB 1022 October 14, 2012
Simplify 6 decimal polynomials with a single variable 747.7 kB 706 October 14, 2012
Simplify 7 decimal polynomials with a single variable 1.2 MB 808 October 14, 2012
Simplify 8 decimal polynomials with a single variable 1.3 MB 635 October 14, 2012
Simplify 9 decimal polynomials with a single variable 1.3 MB 706 October 14, 2012
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 4.2 MB 810 October 14, 2012
Simplify 5 polynomial fractions with a single variable 11.7 MB 707 October 14, 2012
Simplify 6 polynomial fractions with a single variable 1.3 MB 751 October 14, 2012
Simplify 7 polynomial fractions with a single variable 2.7 MB 624 October 14, 2012
Simplify 8 polynomial fractions with a single variable 1.9 MB 826 October 14, 2012
Simplify 8 polynomial fractions with a single variable 2.1 MB 685 October 14, 2012
Two variables – Integers
Simplify 4 integer polynomials with two variables 1 MB 1544 October 14, 2012
Simplify 5 integer polynomials with two variables 1.2 MB 983 October 14, 2012
Simplify 6 integer polynomials with two variables 1.3 MB 876 October 14, 2012
Simplify 7 integer polynomials with two variables 1.4 MB 762 October 14, 2012
Simplify 8 integer polynomials with two variables 1.5 MB 933 October 14, 2012
Simplify 9 integer polynomials with two variables 1.6 MB 893 October 14, 2012
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 1.1 MB 720 October 14, 2012
Simplify 5 decimal polynomials with two variables 1.2 MB 764 October 14, 2012
Simplify 6 decimal polynomials with two variables 1.4 MB 832 October 14, 2012
Simplify 7 decimal polynomials with two variables 1.4 MB 613 October 14, 2012
Simplify 8 decimal polynomials with two variables 1.5 MB 616 October 14, 2012
Simplify 9 decimal polynomials with two variables 1.6 MB 641 October 14, 2012
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 10.1 MB 877 October 14, 2012
Simplify 5 polynomial fractions with two variables 4.2 MB 772 October 14, 2012
Simplify 6 polynomial fractions with two variables 1.3 MB 613 March 5, 2012
Simplify 7 polynomial fractions with two variables 2 MB 623 October 14, 2012
Simplify 8 polynomial fractions with two variables 2.2 MB 778 October 14, 2012
Simplify 9 polynomial fractions with two variables 2.3 MB 821 October 14, 2012