mostafa essa
Admin
عدد المساهمات : 158
تاريخ التسجيل : 26/10/2010
 | موضوع: 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:
| |
|
