# 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 4624 September 3, 2019 Integer polynomials with a single variable – easy 462.6 kB 2842 September 3, 2019 Integer polynomials with a single variable – medium 462.5 kB 3574 September 3, 2019 Integer polynomials with a single variable – hard 465.2 kB 2995 September 3, 2019 Integer polynomials with a single variable – very hard 470.8 kB 2869 September 3, 2019 Single variable – Decimals Decimal polynomials with a single variable – very easy 461.9 kB 1901 September 3, 2019 Decimal polynomials with a single variable – easy 462.3 kB 1773 September 3, 2019 Decimal polynomials with a single variable – medium 461.9 kB 1757 September 3, 2019 Decimal polynomials with a single variable – hard 465.5 kB 1682 September 3, 2019 Decimal polynomials with a single variable – very hard 468.7 kB 1718 September 3, 2019 Single variable – Fractions Polynomial fractions with a single variable – very easy 468.3 kB 1920 September 3, 2019 Polynomial fractions with a single variable – easy 466.9 kB 2007 September 3, 2019 Polynomial fractions with a single variable – medium 468.1 kB 1801 September 3, 2019 Polynomial fractions with a single variable – hard 473.7 kB 2018 September 3, 2019 Polynomial fractions with a single variable – very hard 481.4 kB 2053 September 3, 2019 Two variables – Integers Integer polynomials with two variables – very easy 464.7 kB 2295 September 3, 2019 Integer polynomials with two variables – easy 464.6 kB 2123 September 3, 2019 Integer polynomials with two variables – medium 465 kB 2523 September 3, 2019 Integer polynomials with two variables – hard 469.3 kB 2136 September 3, 2019 Integer polynomials with two variables – very hard 474.1 kB 2266 September 3, 2019 Two variables – Decimals Decimal polynomials with two variables – very easy 465 kB 1631 September 3, 2019 Decimal polynomials with two variables – easy 465.6 kB 1541 September 3, 2019 Decimal polynomials with two variables – medium 465 kB 1713 September 3, 2019 Decimal polynomials with two variables – hard 470.5 kB 1635 September 3, 2019 Decimal polynomials with two variables – very hard 475.9 kB 1722 September 3, 2019 Two variables – Fractions Polynomial fractions with two variables – very easy 470.3 kB 1863 September 3, 2019 Polynomial fractions with two variables – easy 470.4 kB 1560 September 3, 2019 Polynomial fractions with two variables – medium 471.1 kB 1520 September 3, 2019 Polynomial fractions with two variables – hard 479.3 kB 1591 September 3, 2019 Polynomial fractions with two variables – very hard 487.3 kB 1698 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 3861 September 3, 2019 Simplify 5 integer polynomials with a single variable 996.2 kB 2133 September 3, 2019 Simplify 6 integer polynomials with a single variable 1 MB 2088 September 3, 2019 Simplify 7 integer polynomials with a single variable 1.2 MB 1886 September 3, 2019 Simplify 8 integer polynomials with a single variable 1.2 MB 1767 September 3, 2019 Simplify 9 integer polynomials with a single variable 1.3 MB 2762 September 3, 2019 Single variable – Decimals Simplify 4 decimal polynomials with a single variable 927.6 kB 1980 September 3, 2019 Simplify 5 decimal polynomials with a single variable 1 MB 1850 September 3, 2019 Simplify 6 decimal polynomials with a single variable 747.7 kB 1494 September 3, 2019 Simplify 7 decimal polynomials with a single variable 1.2 MB 1940 September 3, 2019 Simplify 8 decimal polynomials with a single variable 1.3 MB 1571 September 3, 2019 Simplify 9 decimal polynomials with a single variable 1.3 MB 1758 September 3, 2019 Single variable – Fractions Simplify 4 polynomial fractions with a single variable 4.2 MB 1836 September 3, 2019 Simplify 5 polynomial fractions with a single variable 11.7 MB 1543 September 3, 2019 Simplify 6 polynomial fractions with a single variable 1.3 MB 1586 September 3, 2019 Simplify 7 polynomial fractions with a single variable 2.7 MB 1583 September 3, 2019 Simplify 8 polynomial fractions with a single variable 1.9 MB 1947 September 3, 2019 Simplify 8 polynomial fractions with a single variable 2.1 MB 1657 September 3, 2019 Two variables – Integers Simplify 4 integer polynomials with two variables 1 MB 4505 September 3, 2019 Simplify 5 integer polynomials with two variables 1.2 MB 2220 September 3, 2019 Simplify 6 integer polynomials with two variables 1.3 MB 2213 September 3, 2019 Simplify 7 integer polynomials with two variables 1.4 MB 1917 September 3, 2019 Simplify 8 integer polynomials with two variables 1.5 MB 2136 September 3, 2019 Simplify 9 integer polynomials with two variables 1.6 MB 2579 September 3, 2019 Two variables – Decimals Simplify 4 decimal polynomials with two variables 1.1 MB 1638 September 3, 2019 Simplify 5 decimal polynomials with two variables 1.2 MB 1619 September 3, 2019 Simplify 6 decimal polynomials with two variables 1.4 MB 1611 September 3, 2019 Simplify 7 decimal polynomials with two variables 1.4 MB 1383 September 3, 2019 Simplify 8 decimal polynomials with two variables 1.5 MB 1524 September 3, 2019 Simplify 9 decimal polynomials with two variables 1.6 MB 1580 September 3, 2019 Two variables – Fractions Simplify 4 polynomial fractions with two variables 10.1 MB 2075 September 3, 2019 Simplify 5 polynomial fractions with two variables 4.2 MB 1799 September 3, 2019 Simplify 6 polynomial fractions with two variables 1.3 MB 1568 September 3, 2019 Simplify 7 polynomial fractions with two variables 2 MB 1439 September 3, 2019 Simplify 8 polynomial fractions with two variables 2.2 MB 1750 September 3, 2019 Simplify 9 polynomial fractions with two variables 2.3 MB 2032 September 3, 2019