Kategorier
Algebra Gymnasiematematik(high school math) matematik 2c

Solving polynomial equations, descartes theorem

A large part of the algebra courses at upper secondary-school level are devoted to solving equations or factorization of polynomials. This is often the same thing.

Some terminology. All of these are the same:
Solving a polynomial equation.
Finding roots of a polynomial equation p(x)=0.
Finding zeros of a polynomial equation p(x)
Factorizing a polynomial function p(x).

There is a factor for every root and vice versa.
(x-r) is a factor if and only if r is a root according to the Factor theorem.

How to solve equations step-by-step:
1. If solving an equation, put it in standard form with 0 on one side and simplif.
2. Know hoewmany roots to expect.
3. Find one factor or root. (Several techniques available)
4. Divide by your factor. This leaves you with a new reduced polynomial. whose degree is 1 less. For the rest of the problem you’ll work with the reduced polynomial and not the original.
5. Now you have a quadratic or linear equation which you already know how to solve.
Write down the solution.

There is no general solution for solving equations of degree 5 and higher.
Try to factorize the polynomial as much as possible.
Equations of degree four or less can be solved by stanard methods.

A plynomial of degree n will have n roots some of which may be multiple roots. This is according to the Fundamental theorem of Algebra.

Descartes Rule of Sign:
Tells you the how many positiv or negative real zeroes the polynomial has.
1. The number of positive roots of p(x)=0 is either equal to the number of variations in sign of p*(x) or less than that by an even numer.
2. The number of negative roots of p(x) = 0 is either equal to the number of variations in sign of p(-x)=0 or less than that by an even number.

Complex Roots

If a polynomial has real coefficients, then either all roots are real or there are an even number of non-real complex roots, in conjugate pairs.

For example, if 5+2i is a zero of a polynomial with real coefficients, then 5−2i must also be a zero of that polynomial. It is equally true that if (x−5−2i) is a factor then (x−5+2i) is also a factor.

This is true because when you have a factor with an imaginary part and multiply it by its complex conjugate you get a real result:

(x−5−2i)(x−5+2i) = x²−10x+25−4i² = x²−10x+29

If (x−5−2i) was a factor but (x−5+2i) was not, then the polynomial would end up with imaginaries in its coefficients, no matter what the other factors might be. If the polynomial has only real coefficients, then any complex roots must occur in conjugate pairs.

Kategorier
Geometri Gymnasiematematik(high school math)

The area of the circle

‘Minute Physics’ derives the area of the circle with a string of pearls and a ruler:

Kategorier
Algebra Gymnasiematematik(high school math)

Modular arithmetic & the Chinese Rest Theorem

modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers ”wrap around” upon reaching a certain value—the modulus.

An example of this is the clock which starts repeating itself when it has reached 12. Therefore it is said that the clock follows an arithmetic modulo 12.  

Ex: If you add 7 hours to 8 o’clock the result is not 15 since when thee clock reaches zero it starts over from zero once again so the result is 7+8=3 in this arithmetic.

For a positive integer n, two integers a and b are said to be congruent modulo n,
a=b mod(n)
if the the remainder is the same when a and b are divided by n.

if their difference a − b is an integer multiple of n. The number n is called the modulus of the congruence.

a \equiv b \pmod n,\, 
e.g. 14

14≡12 (mod 2)

An example of an application of this mathematical tool is the Chinese Remainder Theorem.

Kategorier
Gymnasiematematik(high school math)

Exceldiagrams

Excel is a versatile programme for e.g. statisitics.

Learn more!

Kategorier
Astronomy Uncategorized

Asteroid-passage

In the evening today February the 15th. an asteroid flies by close to the Earth only about 27 600 km away. It measures about 40 m in length and is similar to the one responsible for the Tunguska (Siberia) impact in  1908. It deforestated everything within 30 km.

NASA provides more information.

Yesterday meteorites impacted Russia in the Tjeljabinsk-region injuring 200 people.

Kategorier
Geometri Gymnasiematematik(high school math) matematik 2c

Conical sections

280px-Conic_section_-_clean

The force of gravity determines the trajectories of the celestial bodies. Mathematical analysis reveals that there are three types of trajectories possible for a body moving in a gravitational field determined by Newton’s law of gravity.

  • If one of the bodies has very high speed relative to the other the moving body traces out a hyperbola. The equation for this is
    x 2/a2 – y 2/b=1
  • If the speeds of the bodies are beelow a certain threshold value they move in an elliptical curve. x 2/a2 + y 2/b2 =1
  • The limiting case between the hyperbola and the ellipse is the parabola. y=ax

Those geometrical objects can be illustrated by slicing the cone according to the figure above.