Verify the relationship between zeroes and coefficients of the cubic polynomial p(x) = 3x^{3} - 5x^{2} - 11x - 3p(x)=3x3−5x2−11x−3.

Asked by bibhudeepanp | 1st Jul, 2021, 10:03: PM

Expert Answer:

P(x) = 3 x3 - 5 x2 -11 x -3   ............................. (1)
 
If we put x = -1 , we get P(-1) = 0 
 
Hence (x+1) is a factor of P(x)
 
After dividing P(x) by (x+1) , we get , P(x) = (x+1) ( 3x2 -8x -3 ) 
 
After factorising ( 3x-8x -3 ) , we get,  P(x) = (x+1) (x-3)(3x+1)
 
Cubic polynomial with zeros of polynomials as -a, -b and -c is given as
 
(x+a) (x+b) ( x+c) = x3 + ( a +b +c )x2 + ( ab + bc + ca ) x + (abc) ......................(2)
 
Given polynomial P(x) can be written in terms of factors as 
 
P(x) = (1/3) ( x+1) ( x-3 ) [ x + (1/3) ] = (1/3) [ x3 - (5/3)x2 -(11/3)x -1 ] ..................(3)
 
we get from the factors of polynomial ,  a = 1  , b = -3  and c = (1/3)
 
By comparing eqn.(2) and (3) , we should get , a+b+c = -5/3 . This can be verified by substituting values of a, b and c 
 
Similarly ( ab+bc+ca ) = -5/3 . This can be verified by substituting values of a, b and c
 
Similarly  ( a b c ) = 1 × (-3) × (1/3) = -1
 

 

Answered by Thiyagarajan K | 18th Sep, 2021, 03:49: PM