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 1970 October 13, 2012
Integer polynomials with a single variable – easy 462.6 kB 1044 October 13, 2012
Integer polynomials with a single variable – medium 462.5 kB 1366 October 13, 2012
Integer polynomials with a single variable – hard 465.2 kB 1272 October 13, 2012
Integer polynomials with a single variable – very hard 470.8 kB 1212 October 13, 2012
Single variable – Decimals
Decimal polynomials with a single variable – very easy 461.9 kB 752 October 13, 2012
Decimal polynomials with a single variable – easy 462.3 kB 694 October 13, 2012
Decimal polynomials with a single variable – medium 461.9 kB 733 October 13, 2012
Decimal polynomials with a single variable – hard 465.5 kB 685 October 13, 2012
Decimal polynomials with a single variable – very hard 468.7 kB 700 October 13, 2012
Single variable – Fractions
Polynomial fractions with a single variable – very easy 468.3 kB 848 October 13, 2012
Polynomial fractions with a single variable – easy 466.9 kB 755 October 13, 2012
Polynomial fractions with a single variable – medium 468.1 kB 781 October 13, 2012
Polynomial fractions with a single variable – hard 473.7 kB 787 October 13, 2012
Polynomial fractions with a single variable – very hard 481.4 kB 790 October 13, 2012
Two variables – Integers
Integer polynomials with two variables – very easy 464.7 kB 962 October 13, 2012
Integer polynomials with two variables – easy 464.6 kB 850 October 13, 2012
Integer polynomials with two variables – medium 465 kB 1000 October 13, 2012
Integer polynomials with two variables – hard 469.3 kB 980 October 13, 2012
Integer polynomials with two variables – very hard 474.1 kB 890 October 13, 2012
Two variables – Decimals
Decimal polynomials with two variables – very easy 465 kB 619 October 13, 2012
Decimal polynomials with two variables – easy 465.6 kB 675 October 13, 2012
Decimal polynomials with two variables – medium 465 kB 687 October 13, 2012
Decimal polynomials with two variables – hard 470.5 kB 669 October 13, 2012
Decimal polynomials with two variables – very hard 475.9 kB 792 October 13, 2012
Two variables – Fractions
Polynomial fractions with two variables – very easy 470.3 kB 685 October 13, 2012
Polynomial fractions with two variables – easy 470.4 kB 637 October 13, 2012
Polynomial fractions with two variables – medium 471.1 kB 680 October 13, 2012
Polynomial fractions with two variables – hard 479.3 kB 649 October 13, 2012
Polynomial fractions with two variables – very hard 487.3 kB 736 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 1582 October 14, 2012
Simplify 5 integer polynomials with a single variable 996.2 kB 836 October 14, 2012
Simplify 6 integer polynomials with a single variable 1 MB 906 October 14, 2012
Simplify 7 integer polynomials with a single variable 1.2 MB 866 October 14, 2012
Simplify 8 integer polynomials with a single variable 1.2 MB 800 October 14, 2012
Simplify 9 integer polynomials with a single variable 1.3 MB 1373 October 14, 2012
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 927.6 kB 881 October 14, 2012
Simplify 5 decimal polynomials with a single variable 1 MB 1058 October 14, 2012
Simplify 6 decimal polynomials with a single variable 747.7 kB 737 October 14, 2012
Simplify 7 decimal polynomials with a single variable 1.2 MB 861 October 14, 2012
Simplify 8 decimal polynomials with a single variable 1.3 MB 668 October 14, 2012
Simplify 9 decimal polynomials with a single variable 1.3 MB 754 October 14, 2012
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 4.2 MB 860 October 14, 2012
Simplify 5 polynomial fractions with a single variable 11.7 MB 746 October 14, 2012
Simplify 6 polynomial fractions with a single variable 1.3 MB 791 October 14, 2012
Simplify 7 polynomial fractions with a single variable 2.7 MB 656 October 14, 2012
Simplify 8 polynomial fractions with a single variable 1.9 MB 888 October 14, 2012
Simplify 8 polynomial fractions with a single variable 2.1 MB 718 October 14, 2012
Two variables – Integers
Simplify 4 integer polynomials with two variables 1 MB 1628 October 14, 2012
Simplify 5 integer polynomials with two variables 1.2 MB 1043 October 14, 2012
Simplify 6 integer polynomials with two variables 1.3 MB 927 October 14, 2012
Simplify 7 integer polynomials with two variables 1.4 MB 808 October 14, 2012
Simplify 8 integer polynomials with two variables 1.5 MB 979 October 14, 2012
Simplify 9 integer polynomials with two variables 1.6 MB 949 October 14, 2012
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 1.1 MB 759 October 14, 2012
Simplify 5 decimal polynomials with two variables 1.2 MB 817 October 14, 2012
Simplify 6 decimal polynomials with two variables 1.4 MB 873 October 14, 2012
Simplify 7 decimal polynomials with two variables 1.4 MB 646 October 14, 2012
Simplify 8 decimal polynomials with two variables 1.5 MB 653 October 14, 2012
Simplify 9 decimal polynomials with two variables 1.6 MB 678 October 14, 2012
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 10.1 MB 921 October 14, 2012
Simplify 5 polynomial fractions with two variables 4.2 MB 822 October 14, 2012
Simplify 6 polynomial fractions with two variables 1.3 MB 645 March 5, 2012
Simplify 7 polynomial fractions with two variables 2 MB 662 October 14, 2012
Simplify 8 polynomial fractions with two variables 2.2 MB 836 October 14, 2012
Simplify 9 polynomial fractions with two variables 2.3 MB 875 October 14, 2012