Question

if two zeroes of the polynomial f(x)=x3-4x2-3x+12 are √3 and -√3 then find its third zero

Answer 1

Heya !!!




Given that :-


root 3 and - root 3 are the two zeroes of the polynomial X³-4X²-3X+12 .





(X - root 3 ) ( X + root 3 ) is also the factor of the given polynomial.




Therefore,



( X - root 3) ( X + root 3) = (X)² - (root 3)²




=> (X² - 3) is a factor of the given polynomial.





P(X) = X³ - 4X² - 3X + 12



G(X) = X² -3




On dividing P(X) by G(X) we get:




X² - 3 ) X³ - 4X² - 3X + 12 ( X -4


*********X³********-3X

--------------------------------------

******0*****-4X²******+12


************-4X²*******+12

---------------------------------------

Remainder = 0





We get,



Remainder = 0

And,


Quotient = X - 4




=> X -4 = 0



=> X = 4




Hence,

third zero of the given polynomial is 4.




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Answer 2

Step-by-step explanation:

f(x)=x3-4x2-3x+12

α=+√3, β=-√3, γ=?

using relation between zero and coefficient:

αβγ=-d/a

+√3 x -√3 x γ = -12/1 = -12

-3 x γ = -12

γ= -12/-3

 = 4

==> the third zero is 4.