Polynomial

Definition of a polynomial

Before giving you the definition of a polynomial, it is important to provide the definition of a monomial

Definition of a monomial:

A monomial is a variable, a real number, or a multiplication of one or more variables and a real number with whole-number exponents

Examples of monomials and non-monomials

Monomials	9	x	9x	6xy	0.60x4y
Not monomials	y - 6	x-1 or 1/x	√(x) or x1/2	6 + x	a/x

Polynomial definition:

A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the poynomial

Important!:Terms are seperated by addition signs and subtraction signs, but never by multiplication signs

A polynomial with one term is called a monomial

A polynomial with two terms is called a binomial

A polynomial with three terms is called a trinomial


Examples of polynomials:

Polynomial	Number of terms	Some examples
Monomial	1	2, x, 5x3
Binomial	2	2x + 5, x2 - x, x - 5
Trinomial	3	x2 + 5x + 6, x5 - 3x + 8

Difference between a monomial and a polynomial:

A polynomial may have more than one variable.

For example, x + y and x² + 5y + 6 are still polynomials although they have two different variables x and y

By the same token, a monomial can have more than one variable. For example, 2 × x × y × z is a monomial



Exercices

For all expressions below, look for all expressions that are polynomials.

For those that are polynomials, state whether the polynomial is a monomial, a binomial, or a trinomial

1) 3.4 + 3.4x

2) z² + 5z^-1 + 6

3) -8

4) 2c² + 5b + 6

5) 14 + x

6) 5x - 2^-1

7) 4 b² - 2 b^-2

8) f² + 5f + 6

Answer: 1), 3), 4), 5), 6), and 8) are polynomials. 1), 5), and 6) are binomials. 3) is a monomial. 4) and 8) are trinomials

2) and 7) are not because they have negative exponents

Notice that 6) is still a polynomial although it has a negative exponent. It is because it is the exponent of a real number, not a variable

In fact, 5x - 2^-1 = 5x + 1/2 = 5x + 0.5

It is subtle, but if you have any questions about the definition of a polynomial, feel free to contact me

what is the difference between differentiate and derivate?

Differentiation is a process. Derivative is the result obtained from differentiation.

y=2x3

dydx=6x2

is the derivative obtained by differentiation.

differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables relating:

• The values of the function itself, and
• Its derivatives of various orders.

Order

The order of a differential equation is defined as being that of the highest order derivative that is present in the equation.

ODE and PDE

Ordinary Differential Equation

An ordinary differential equation (abbreviated O.D.E. or ODE) is one which has only one independent variable.

All the derivatives occurring in it are therefore ordinary. 

Partial Differential Equation

A partial differential equation (abbreviated P.D.E. or PDE) is one which has more than one independent variable.

The derivatives occurring in it are therefore partial.

linear and non linear

Linear

A linear differential equation is one where any dependent variables and their derivatives appear to the first power.
Neither are products of dependent variables allowed.

Non-Linear

A non-linear differential equation is one which is not linear.

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hyperbola

A curve shaped like an arch.

The distances of any point on a hyperbola from ...

... a fixed point (the focus), and ...
... a fixed straight line (the directrix) ...

... are always in the same ratio.

It is one of the "Conic Sections"