Question

Obtain all zeroes of the polynomial 2x⁴ – 2x³ –7x² + 3x + 6 if two factors of this polynomial are ( x ± √3 /2) .

Answer 1

Method of finding the remaining zeros of a polynomial when sum of  its zeros are given:
We firstly write the factor of polynomial using given zeros and multiply them to get g(x). Then divide a given polynomial by g(x).
The quotient so obtained give other zeros of given polynomial and we factorise it to get other zeros.
SOLUTION:
Let f(x) = 2x⁴  – 2x³  –7x²  + 3x  + 6
Given :
(x+√3/2) & (x-√3/2) are the two factors of given Polynomial f(x).

(x+√3/2) (x-√3/2)
= x²- 3/2 = (2x²-3)/2
= (2x²-3)/2=0
2x²-3 is a factor of given Polynomial f(x)
Divide f(x) =2x⁴  – 2x³  –7x²  + 3x  + 6 by 2x²-3
[DIVISION IS IN THE ATTACHMENT.]

f(x) =2x⁴  – 2x³  –7x²  + 3x  + 6
= (2x²-3)(x²-x-2)
= (2x²-3) (x-2) (x-1)

f(x)=0
= (2x²-3) =0 , (x-2)= 0 , (x-1)=0
x= √3/2, -√3/2, 2 ,-1

Hence , all the zeroes of the given Polynomial are: (√3/2), (-√3/2), -1,2
HOPE THIS WILL HELP YOU …..

Answer 2

all zeros of polynomial is
$\sqrt{3 } \div 2 \: and \: - \sqrt{3} \div 2$
and -1, 2