Question

\(\Large{Tutorial:-}\)


How to find the quadratic & cubic polynomials if there zeroes are given .....

Answer 1

QUADRATIC polynomial :-
If zeroes are α & β then polynomial will be find by this :-
\(x^2−x(α+β)+αβ\)


CUBIC polynomial :-
If zeroes are α,β & γ then polynomial will be find by this :-
\(x^3−x^2(α+β+γ)+x(αβ+βγ+γα)−αβγ\)

Answer 2

Example1:- Find a quadratic polynomial, the sum & product of whose zeroes are -5 & 6 respectively.
Solution:- Let \(\alpha\ \&\ \beta\) are the zeroes of the required polynomial \(f(x)\).
Then, \(\alpha+\beta\)=-5 and \(\alpha \beta\)=6.
\(\dot{\ddot{}}\)\(f(x)=x^2-x(\alpha+\beta )+\alpha \beta\)
\(\implies f(x)=x^2-(-5x)+6\)
\(\implies f(x)=x^2+5x+6.\)
Hence, the required polynomial is \(f(x)=x^2+5x+6\)

Answer 3

Example2 :- Find a cubic polynomial whose zeroes are \(\alpha,\beta \) & \(\gamma\) such that \((\alpha+\beta+\gamma)\)=6, \((\alpha\beta+\beta\gamma+\gamma\alpha)\)=-1 & \((\alpha\beta\gamma)\)=-30.
Solution :- We know that a cubic polynomial whose zeroes are \(\alpha\beta\) & \(\gamma\) is given by
\(p(x)=x^3-x^2(\alpha+\beta+\gamma)+x(\alpha\beta+\beta\gamma+\gamma\alpha)\)-\(\alpha\beta\gamma\)
\(\implies p(x)=x^3-6x^2-x-(-30)\)
\(\implies p(x)=x^3-6x^2-x+30\)
Hence, the required polynomial is \((x^3-6x^2-x+30)\).

Answer 4

@mathslover @vishweshshrimali5 @UnkleRhaukus
I now gave examples as u siad :)

Answer 5

What about a quadratic molynomial structure?

Answer 6

hmm... I m too small for that :D

Answer 7

LOL

Answer 8

One suggestion ........

Answer 9

Fast

Answer 10

Please increase the size of the latex

Answer 11

Same was given by @sasogeek :D

Answer 12

:D

Answer 13

LOL it is for help :P

Answer 14

\[The \ Viper \\ \huge\ GOOD \ WORK\]

Answer 15

\[\Huge{\color{blue}{\cal{Thanx}}\color{green}{\ddot{\smile}}}\]@Rohangrr