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 4966 September 3, 2019
Integer polynomials with a single variable – easy 462.6 kB 3132 September 3, 2019
Integer polynomials with a single variable – medium 462.5 kB 3869 September 3, 2019
Integer polynomials with a single variable – hard 465.2 kB 3299 September 3, 2019
Integer polynomials with a single variable – very hard 470.8 kB 3145 September 3, 2019
Single variable – Decimals
Decimal polynomials with a single variable – very easy 461.9 kB 2144 September 3, 2019
Decimal polynomials with a single variable – easy 462.3 kB 2009 September 3, 2019
Decimal polynomials with a single variable – medium 461.9 kB 1989 September 3, 2019
Decimal polynomials with a single variable – hard 465.5 kB 1940 September 3, 2019
Decimal polynomials with a single variable – very hard 468.7 kB 1967 September 3, 2019
Single variable – Fractions
Polynomial fractions with a single variable – very easy 468.3 kB 2162 September 3, 2019
Polynomial fractions with a single variable – easy 466.9 kB 2227 September 3, 2019
Polynomial fractions with a single variable – medium 468.1 kB 2047 September 3, 2019
Polynomial fractions with a single variable – hard 473.7 kB 2238 September 3, 2019
Polynomial fractions with a single variable – very hard 481.4 kB 2276 September 3, 2019
Two variables – Integers
Integer polynomials with two variables – very easy 464.7 kB 2568 September 3, 2019
Integer polynomials with two variables – easy 464.6 kB 2377 September 3, 2019
Integer polynomials with two variables – medium 465 kB 2779 September 3, 2019
Integer polynomials with two variables – hard 469.3 kB 2402 September 3, 2019
Integer polynomials with two variables – very hard 474.1 kB 2529 September 3, 2019
Two variables – Decimals
Decimal polynomials with two variables – very easy 465 kB 1861 September 3, 2019
Decimal polynomials with two variables – easy 465.6 kB 1771 September 3, 2019
Decimal polynomials with two variables – medium 465 kB 1984 September 3, 2019
Decimal polynomials with two variables – hard 470.5 kB 1871 September 3, 2019
Decimal polynomials with two variables – very hard 475.9 kB 2081 September 3, 2019
Two variables – Fractions
Polynomial fractions with two variables – very easy 470.3 kB 2073 September 3, 2019
Polynomial fractions with two variables – easy 470.4 kB 1771 September 3, 2019
Polynomial fractions with two variables – medium 471.1 kB 1745 September 3, 2019
Polynomial fractions with two variables – hard 479.3 kB 1790 September 3, 2019
Polynomial fractions with two variables – very hard 487.3 kB 1903 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 4211 September 3, 2019
Simplify 5 integer polynomials with a single variable 996.2 kB 2468 September 3, 2019
Simplify 6 integer polynomials with a single variable 1 MB 2355 September 3, 2019
Simplify 7 integer polynomials with a single variable 1.2 MB 2210 September 3, 2019
Simplify 8 integer polynomials with a single variable 1.2 MB 2064 September 3, 2019
Simplify 9 integer polynomials with a single variable 1.3 MB 3015 September 3, 2019
Single variable – Decimals
Simplify 4 decimal polynomials with a single variable 927.6 kB 2199 September 3, 2019
Simplify 5 decimal polynomials with a single variable 1 MB 2093 September 3, 2019
Simplify 6 decimal polynomials with a single variable 747.7 kB 1707 September 3, 2019
Simplify 7 decimal polynomials with a single variable 1.2 MB 2149 September 3, 2019
Simplify 8 decimal polynomials with a single variable 1.3 MB 1801 September 3, 2019
Simplify 9 decimal polynomials with a single variable 1.3 MB 1966 September 3, 2019
Single variable – Fractions
Simplify 4 polynomial fractions with a single variable 4.2 MB 2045 September 3, 2019
Simplify 5 polynomial fractions with a single variable 11.7 MB 1767 September 3, 2019
Simplify 6 polynomial fractions with a single variable 1.3 MB 1799 September 3, 2019
Simplify 7 polynomial fractions with a single variable 2.7 MB 1800 September 3, 2019
Simplify 8 polynomial fractions with a single variable 1.9 MB 2165 September 3, 2019
Simplify 8 polynomial fractions with a single variable 2.1 MB 1897 September 3, 2019
Two variables – Integers
Simplify 4 integer polynomials with two variables 1 MB 5025 September 3, 2019
Simplify 5 integer polynomials with two variables 1.2 MB 2821 September 3, 2019
Simplify 6 integer polynomials with two variables 1.3 MB 2789 September 3, 2019
Simplify 7 integer polynomials with two variables 1.4 MB 2133 September 3, 2019
Simplify 8 integer polynomials with two variables 1.5 MB 2723 September 3, 2019
Simplify 9 integer polynomials with two variables 1.6 MB 3039 September 3, 2019
Two variables – Decimals
Simplify 4 decimal polynomials with two variables 1.1 MB 1843 September 3, 2019
Simplify 5 decimal polynomials with two variables 1.2 MB 1839 September 3, 2019
Simplify 6 decimal polynomials with two variables 1.4 MB 1832 September 3, 2019
Simplify 7 decimal polynomials with two variables 1.4 MB 1590 September 3, 2019
Simplify 8 decimal polynomials with two variables 1.5 MB 1750 September 3, 2019
Simplify 9 decimal polynomials with two variables 1.6 MB 1800 September 3, 2019
Two variables – Fractions
Simplify 4 polynomial fractions with two variables 10.1 MB 2292 September 3, 2019
Simplify 5 polynomial fractions with two variables 4.2 MB 2014 September 3, 2019
Simplify 6 polynomial fractions with two variables 1.3 MB 1774 September 3, 2019
Simplify 7 polynomial fractions with two variables 2 MB 1658 September 3, 2019
Simplify 8 polynomial fractions with two variables 2.2 MB 1966 September 3, 2019
Simplify 9 polynomial fractions with two variables 2.3 MB 2244 September 3, 2019