Sum and Product of Roots | Symmetrical Functions of root

By symmetric function of roots, we mean that the function remains unchanged when the roots are interchanged.

Let α and β  be the roots.
A function in α  and β is said to be a symmetric function if  the function remains the same when α and β are interchanged.

Consider α+β.  When α is written as β, and  β is written as α, the function becomes β+α, which is same as α+β.

So α+β is a symmetric function of the roots.
αβ is also a symmetric function of the roots as αβ = βα

α-β is not a symmetric function as α-β is not equal to β-α unless if β=α

However, mod(α-β) is a symmetric function, as mod(α-β )= mod(β-α)

Value of a #symmetric #function of the #roots in terms of the coefficients of the quadratic equation

α +β = -b/a
αβ = c/a

Value of a symmetric function of α and β can be obtained if α+β and αβ are known.

To find the value of the symmetric function of the roots, express the given function in terms of α+β and αβ.

You can use the following results:

 α²+β²   = (α +β)² - 2αβ

 α³+β³  = (α +β)³ - 3αβ(α+β)

 α⁴+β⁴ = (α³+β³)(α +β)-αβ(α²+β²)

(α+β)⁴= α⁴+6α³β+4α²β²+6αβ³+β⁴ 
= α⁴+β⁴+6αβ(α²+β²)+4α²β²
   
α⁵+β⁵ = (α³+β³)(α²+β²)-(α²β³+α³β²)
=(α³+β³)(α²+β²)-(α²β²)(α+β)

mod(α -β ) = √[(α +β)²-4αβ]

α²-β²= (α+β)(α-β)

α³-β³ = (α-β)³+3αβ(α-β)
= (α-β)[(α +β)²-αβ]

α⁴-β⁴ = (α²+β²)(α²-β²)
= (α+β)(α-β)(α²+β²)

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[francisnjie]

I appreciate your work deeply sir, it's realy been helpful. Please could you make this clear: you (a ^2+ B ^2), when you were done expanding, how did a minus sign come before the 2aB ?

[robinsonmugala]

wow....i got what have been looking at...solving sam of pefect roots....

[pipslord]

In the first question...
The third line were the parenthesis was expanded ...
U used -2 instead of +2 ...can u be clear about that?

[mapenzihamoomba]

Great videos sir👏🏾….appreciation from Zambia 🇿🇲😃

[tinodaisheganyani]

Your teaching sir is much understandable .Keep doing such good work

[rahmatbadamasi]

Thanks so much more knowledge. I have really learn a lot jazakhallahu khairan.

[Madepabiayera]

You are a Good teacher
You teaches well❤

[TiwaElise]

How do we make the product of the second problem

[onileyanomotoyosi]

Thank you for your teachings

Allah bless you

Please do you know any study group for secondary school, science students

Thank you

[binibrahim]

Assalam Alaikum warahmatullah.
Masha Allah, this is really great and enjoyable tutorial.
I have a question by the way.
In solving for the 3rd question, can we just make it simple by substituting the values in the first step where it says
(á+ɓ)(á+ɓ)²

[samirzain]

Mashallah tnx bro I really understood well

[codewithshedrack]

This is great... Keep up the good work sir

[Nsikak-abasiEbong-extk]

Please 🙏🥺 help me to solve the root product of question b



Cause it's only the sum product u solved

[cosmasmali]

Can you please express alpha squared minus beta squared????

[mukhtarullah]

Sir please help me lot of because my chemistry exame is near solve this question underoot (x+b/x-b)find Dy/Dx

[maryojo]

Pls can you make a video on cubic root

[PF_BUCHI]

Please can you solve for alpha minus beta