What is the correct definition of polynomial

polynomial

Here you can find out what a polynomial is and what properties it has. You can also find many Examples and an understandable video on the subject.

What is a polynomial

The term polynomial sounds complicated at first, but in fact it only describes a certain type multi-part terms. Examples of this are around

a + b + c

10x5-x + 3x3+4

3x2-8x + 2

A polynomial consists of variablesbetween which there are plus or minus signs. The variables often have

  • one Factor, that is, a number by which the variable is multiplied. He stands in front of the variable.
  • an exponent or power, which is a natural number.

With a polynomial, you actually only add power functions together or subtract them from each other.

Properties of Polynomials

You can quickly recognize the individual parts of a polynomial and its properties from this example:

x2+ 3x-1

variables

Variables are described with letters. Most of the time it's x or x1 , x2 , x3 etc.

  • The polynomial (x2+3x-1) contains only one variable: x

Prefactors / coefficients

Pre-factors or coefficients are the numbers that come before a variable. You always have to multiply the variable by the prefactor.

  • The prefactors of the polynomial (1x2+3x-1) are 1 , 3 and -1 
  • If there is no prefactor in front of a variable (as with x2), then you have to think of a 1, because 1 *x2=x2
  • Pre-factors are also called coefficients

Potencies

Powers are the exponents above the variables. You multiply a variable by itself as often as its exponent indicates. x2 so is x * x.

  • The powers of the polynomial (x2+ 3x1-1x0) are 2,  1 and 0
  • Is there a Variable with no power there (as with 3x), then you have to get one 1 think as a power to it (because x1= x)
  • Stand up Prefactor without variable there (as with -1), then you have to have a power as a 0 think about it (because x0=1)

Limbs

The individual parts of the sum are called terms.

  • x2, 3x and -1are the terms of the term (x2+3x1).

No Polynomials are all more complicated terms such as root or contain fractions whose denominator consists of a variable (broken rational functions).

  • .
Polynomials are often written down sorted according to their exponents. Such polynomials are also called ordered polynomials. This is how you write, for example instead of .

Degree of a polynomial

The degree of a polynomial is always the highest power of the polynomial. So it is the exponent of a variable that is greatest. Here are some examples of the degree of different polynomials:

  • 4x3+ 2x2+ 3x-1 3rd degree polynomial (because of 4x3)
  • -7x5-2x3+12 5th degree polynomial (because of -7x5
  • -7x-2x2+12 2nd degree polynomial (because of 2x2)

Special polynomials

There are different types of polynomials, some of which are particularly important. The best known polynomials are the zero polynomial, the binomial and the trinomial. We will briefly introduce them to you here.

General polynomial

Some of the most important polynomials are those with only one variable x. They are also known as “polynomials in a variable x”. Generally written down they look like this:

Examples of polynomials in a variable x

  • -2x3+ 12x2+ 3x-1
  • 8x5-2x3
  • x8+ 3x5-16x2

Zero polynomial

The simplest possible polynomial is called Zero polynomial. It's constant zero

f (x) = 0.

The degree of the zero polynomial is at -1 or fixed. Its function graph is identical to the x-axis, i.e. it is a horizontal straight line in the coordinate system with a y-axis intercept

Monom

A monom consists of only one term. Examples for this are:

  • a

Binomial

A binomial always consists of two limbs, it is the total or the difference two monomials. Typical examples of such a binomial are

  • a + b
  • 2x2+ x
  • 4ab-b.

Danger: Be careful not to confuse the binomial with the binomial formulas! Here two binomials are multiplied!

Trinome

In contrast to the binomial, a trinomial consists of three individual terms or three monomials. Typical examples are

  • a + b + c
  • 2x + 3-y
  • ax2+ bx + c
  • a2+ 2ab + b2.

Polynomial division

You can also divide polynomials into one another. In principle, this works in the same way as the written sharing in elementary school. Would you like the polynomial 5x2+ 3x-12 divide by the other polynomial x-4, then the whole thing would look like this:

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The Polynomial division you absolutely have to be able to determine zeros of polynomials of the third degree or higher. Of course, we explain in detail how to do this in a separate video. Check it out right now!