منتدى لكل المهندسين
 
الرئيسيةالبوابةبحـثالتسجيلدخول

شاطر | 
 

 شرح مبسط لنظرية ذات الحدين ( جبر رياضة 1)

استعرض الموضوع السابق استعرض الموضوع التالي اذهب الى الأسفل 
كاتب الموضوعرسالة
mostafa essa
Admin
avatar

عدد المساهمات : 158
تاريخ التسجيل : 26/10/2010

مُساهمةموضوع: شرح مبسط لنظرية ذات الحدين ( جبر رياضة 1)   الأربعاء سبتمبر 28, 2011 6:16 am

انا ان شاء
الله جايب ليكم شرح كامل لنظرية ذات الحدين وكمان امثله محلوله ل الى لسه
مفهماش وبالتوفيق يا شباب






A binomial is an algebraic
expression containing 2 terms. For example, (x + y) is a
binomial.

We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b +
3ab2 + b3

(a + b)4 = a4 + 4a3b +
6a2b2 + 4ab3 + b4

(a + b)5 = a5 + 5a4b +
10a3b2 + 10a2b3 + 5ab4 + b5



Clearly, doing this by direct multiplication gets quite
tedious and can be rather difficult for larger powers or more
complicated expressions.



LiveMath Solution

Computers can do this for us very easily. Let's get
LiveMath to expand the binomial for us.



Pascal's Triangle

We note that the coefficients (the
numbers in front of each term) follow a pattern. [This was noticed long
before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

You can use this pattern to form the coefficients, rather
than multiply everything out as we did above
.


The Binomial Theorem

We use the binomial theorem to
help us expand binomials to any given power without direct
multiplication. As we have seen, multiplication can be time-consuming or
even not possible in some cases
.
Properties of the Binomial Expansion (a + b)n

  • There are n + 1 terms.
  • The first term is an and the
    final term is bn.



  • Progressing from the first
    term to the last, the exponent of a decreases by 1 from term to
    term while the exponent of b increases by 1. In addition, the sum
    of the exponents of a and b in each term is n.



  • If the coefficient of each
    term is multiplied by the exponent of a in that term, and the
    product is divided by the number of that term, we obtain the coefficient
    of the next term.

General formula for (a +
b)n



First, we need the following
definition:

Definition: n! represents the product of
the first n positive integers i.e.


<blockquote>
n! = n(n − 1)(n
− 2) ... (3)(2)(1)


</blockquote>We say n! as 'n
factorial
'

The LiveMath expansion of factorials is usually better
than our calculators, in that it can go higher. You can go up to about
170!.



Examples:

3! = (3)(2)(1) = 6
5! = (5)(4)(3)(2)(1) = 120

Note : cannot be cancelled down to 2!


Binomial Theorem Formula


Based on the binomial properties, the binomial theorem
states that the following binomial formula is valid for all
positive integer values of n:
<blockquote>


</blockquote>This can be written more simply as:

We can use the button on our calculator to find these values.
LiveMath can also find just the coefficients [numbers at the
front] for us, too.
These are usually written nCr.


This LiveMath example will find the coefficients nCr.
Your calculator can also do the same thing.



Example 1:

Using the binomial theorem, expand .


Answer

In using the binomial formula, we let
a = x , b = 2 and n = 6.
Substituting in the binomial formula, we get :





Example 2:

Using the binomial theorem, expand


Answer

In using the binomial formula, we let
a = 2x , b = 3 and n = 4.

Substituting in the binomial formula, we get:




Example 3:

Using the binomial theorem, find the first four terms of
the expansion



Answer

In using the binomial formula, we let
a = 2a , b = and n = 11

Substituting in the binomial formula, we get:




Binomial Series

From the binomial formula, if we let a =
1
and b = x, we can also obtain the binomial
series
which is valid for any real number n if |x|
< 1.



NOTE (1): This is an infinite series, where
the binomial theorem deals with a finite expansion.


NOTE (2): We cannot use the button for the binomial series. The button can only be used with positive integers.


Example:

Using the binomial series, find the first four terms of
the expansion .



Answer

In using the binomial series, we need to change
the expansion to the form of


Hence, if we let the x term be and , and
then substituting in the binomial series, we get:






_________________


للقيادة عنوان




by : abo essa....
الرجوع الى أعلى الصفحة اذهب الى الأسفل
 
شرح مبسط لنظرية ذات الحدين ( جبر رياضة 1)
استعرض الموضوع السابق استعرض الموضوع التالي الرجوع الى أعلى الصفحة 
صفحة 1 من اصل 1

صلاحيات هذا المنتدى:لاتستطيع الرد على المواضيع في هذا المنتدى
منتدى هندسه الفيوم :: ( ملتقى الطلاب ) :: إعدادي هندسة-
انتقل الى: