1 What is it all about?
It’s all about XXI century. Would you like to be effective in your Math tests? Would you like to minimize your efforts when dealing with algebra? If the answer is ‘yes’ then read and watch what is presented below.
2 Most useful identities
We begin with the most useful set of binomial identities.
$$ (a^2-b^2) = (a-b)(a+b) \ \ \ \ \ \ \ \ (1) $$
$$ (a+b)^2 = a^2+2ab +b^2 \ \ \ \ \ \ \ \ (2) $$
$$ (a-b)^2 = a^2-2ab + b^2\ \ \ \ \ \ \ (3) $$
In this video (21 min 50 sec) we prove these identities and consider some practical examples. Besides, for those who are strong and curious, in the video we also prove Sophie Germain‘s identity:
$$ a^4 + 4b^4 = (a^2+2ab+ 2b^2)(a^2-2ab+ 2b^2) \ \ \ (4) $$
This beautiful identity is wifely used in difficult olympiad problems.
3 More identities
There are other useful identities. It looks practical to remember the following ones:
$$ a^3 \ + \ b^3 = (a+ b)(a^2-ab+ b^2) \ \ \ \ \ (5) $$
$$ a^3 \ – \ b^3 = (a- b)(a^2+ab+ b^2) \ \ \ \ \ \ (6) $$
(5) is called sum of cubes, and (6) is called difference of cubes.
There are many more beautiful and more complicated identities, but from the prectical point of view it’s a good idea to remember (1)-(6) and derive all the rest which is needed in any particular case.
4 Binomial formula and Pascal’s triangle
Here we cosider the following expansion, which is often called binomial expansion, or binomial formula:
$$ (a+b)^n=\Bigl( \begin{smallmatrix} n \\ 0 \end{smallmatrix} \Bigr)a^n+ \Bigl( \begin{smallmatrix} n \\ 1 \end{smallmatrix} \Bigr)a^{n-1}b +\Bigl( \begin{smallmatrix} n \\ 2 \end{smallmatrix} \Bigr)a^{n-2}b^2 + … + \Bigl( \begin{smallmatrix} n \\ n-1 \end{smallmatrix} \Bigr)ab^{n-1} + \Bigl( \begin{smallmatrix} n \\ n \end{smallmatrix} \Bigr)b^n \ \ \ \ \ \ \ (7)$$
5 How to check the result of comlicated algebraic calculations
6 What next?
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