Find all the zeroes of the polynomial 2x cube + x square - 6 x - 3, if two of its zeroes are - root 3 and root 3.
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Given polynomial,
2x^3+x^2-6x-3=0
also given that -√3 and √3 are roots of given polynomial.
Then factors of given polynomial are,
(x+√3) and (x-√3).
(x+√3)(x-√3)=x^2-3
Product of factors is also a factor of that polynomial.
Now,we have,
=(2x^3+x^2-6x-3)/(x^2-3)
=2x+1
Other zero of given polynomial wil also be the zero of above polynomial i.e (2x+1).
2x+1=0
x=-1/2
Hence all the zeroes of polynomial (2x^3+x^2-6x-3) are √3,-√3 and -1/2.
2x^3+x^2-6x-3=0
also given that -√3 and √3 are roots of given polynomial.
Then factors of given polynomial are,
(x+√3) and (x-√3).
(x+√3)(x-√3)=x^2-3
Product of factors is also a factor of that polynomial.
Now,we have,
=(2x^3+x^2-6x-3)/(x^2-3)
=2x+1
Other zero of given polynomial wil also be the zero of above polynomial i.e (2x+1).
2x+1=0
x=-1/2
Hence all the zeroes of polynomial (2x^3+x^2-6x-3) are √3,-√3 and -1/2.
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