# 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 4273 September 3, 2019 Integer polynomials with a single variable – easy 462.6 kB 2596 September 3, 2019 Integer polynomials with a single variable – medium 462.5 kB 3311 September 3, 2019 Integer polynomials with a single variable – hard 465.2 kB 2756 September 3, 2019 Integer polynomials with a single variable – very hard 470.8 kB 2636 September 3, 2019 Single variable – Decimals Decimal polynomials with a single variable – very easy 461.9 kB 1718 September 3, 2019 Decimal polynomials with a single variable – easy 462.3 kB 1597 September 3, 2019 Decimal polynomials with a single variable – medium 461.9 kB 1582 September 3, 2019 Decimal polynomials with a single variable – hard 465.5 kB 1507 September 3, 2019 Decimal polynomials with a single variable – very hard 468.7 kB 1545 September 3, 2019 Single variable – Fractions Polynomial fractions with a single variable – very easy 468.3 kB 1726 September 3, 2019 Polynomial fractions with a single variable – easy 466.9 kB 1815 September 3, 2019 Polynomial fractions with a single variable – medium 468.1 kB 1621 September 3, 2019 Polynomial fractions with a single variable – hard 473.7 kB 1851 September 3, 2019 Polynomial fractions with a single variable – very hard 481.4 kB 1886 September 3, 2019 Two variables – Integers Integer polynomials with two variables – very easy 464.7 kB 2062 September 3, 2019 Integer polynomials with two variables – easy 464.6 kB 1923 September 3, 2019 Integer polynomials with two variables – medium 465 kB 2293 September 3, 2019 Integer polynomials with two variables – hard 469.3 kB 1933 September 3, 2019 Integer polynomials with two variables – very hard 474.1 kB 2052 September 3, 2019 Two variables – Decimals Decimal polynomials with two variables – very easy 465 kB 1457 September 3, 2019 Decimal polynomials with two variables – easy 465.6 kB 1357 September 3, 2019 Decimal polynomials with two variables – medium 465 kB 1517 September 3, 2019 Decimal polynomials with two variables – hard 470.5 kB 1467 September 3, 2019 Decimal polynomials with two variables – very hard 475.9 kB 1510 September 3, 2019 Two variables – Fractions Polynomial fractions with two variables – very easy 470.3 kB 1702 September 3, 2019 Polynomial fractions with two variables – easy 470.4 kB 1407 September 3, 2019 Polynomial fractions with two variables – medium 471.1 kB 1359 September 3, 2019 Polynomial fractions with two variables – hard 479.3 kB 1448 September 3, 2019 Polynomial fractions with two variables – very hard 487.3 kB 1532 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 3465 September 3, 2019 Simplify 5 integer polynomials with a single variable 996.2 kB 1837 September 3, 2019 Simplify 6 integer polynomials with a single variable 1 MB 1856 September 3, 2019 Simplify 7 integer polynomials with a single variable 1.2 MB 1608 September 3, 2019 Simplify 8 integer polynomials with a single variable 1.2 MB 1553 September 3, 2019 Simplify 9 integer polynomials with a single variable 1.3 MB 2554 September 3, 2019 Single variable – Decimals Simplify 4 decimal polynomials with a single variable 927.6 kB 1802 September 3, 2019 Simplify 5 decimal polynomials with a single variable 1 MB 1679 September 3, 2019 Simplify 6 decimal polynomials with a single variable 747.7 kB 1314 September 3, 2019 Simplify 7 decimal polynomials with a single variable 1.2 MB 1720 September 3, 2019 Simplify 8 decimal polynomials with a single variable 1.3 MB 1369 September 3, 2019 Simplify 9 decimal polynomials with a single variable 1.3 MB 1588 September 3, 2019 Single variable – Fractions Simplify 4 polynomial fractions with a single variable 4.2 MB 1652 September 3, 2019 Simplify 5 polynomial fractions with a single variable 11.7 MB 1351 September 3, 2019 Simplify 6 polynomial fractions with a single variable 1.3 MB 1422 September 3, 2019 Simplify 7 polynomial fractions with a single variable 2.7 MB 1413 September 3, 2019 Simplify 8 polynomial fractions with a single variable 1.9 MB 1795 September 3, 2019 Simplify 8 polynomial fractions with a single variable 2.1 MB 1486 September 3, 2019 Two variables – Integers Simplify 4 integer polynomials with two variables 1 MB 4203 September 3, 2019 Simplify 5 integer polynomials with two variables 1.2 MB 1920 September 3, 2019 Simplify 6 integer polynomials with two variables 1.3 MB 1987 September 3, 2019 Simplify 7 integer polynomials with two variables 1.4 MB 1754 September 3, 2019 Simplify 8 integer polynomials with two variables 1.5 MB 1828 September 3, 2019 Simplify 9 integer polynomials with two variables 1.6 MB 2233 September 3, 2019 Two variables – Decimals Simplify 4 decimal polynomials with two variables 1.1 MB 1466 September 3, 2019 Simplify 5 decimal polynomials with two variables 1.2 MB 1473 September 3, 2019 Simplify 6 decimal polynomials with two variables 1.4 MB 1437 September 3, 2019 Simplify 7 decimal polynomials with two variables 1.4 MB 1231 September 3, 2019 Simplify 8 decimal polynomials with two variables 1.5 MB 1364 September 3, 2019 Simplify 9 decimal polynomials with two variables 1.6 MB 1424 September 3, 2019 Two variables – Fractions Simplify 4 polynomial fractions with two variables 10.1 MB 1900 September 3, 2019 Simplify 5 polynomial fractions with two variables 4.2 MB 1638 September 3, 2019 Simplify 6 polynomial fractions with two variables 1.3 MB 1414 September 3, 2019 Simplify 7 polynomial fractions with two variables 2 MB 1275 September 3, 2019 Simplify 8 polynomial fractions with two variables 2.2 MB 1602 September 3, 2019 Simplify 9 polynomial fractions with two variables 2.3 MB 1847 September 3, 2019