Verify that 5, –2 and are the zeros of the cubic polynomial p(x) = 3x3 - 10x2– 27x + 10 and verify the relation between its zeros and coefficients

It is given in the question that,

p (x) = 3x3 – 10x2 – 27x + 10


Also, 5, -2 and are the zeros of the given polynomial


p (5) = 3 (5)3 – 10 (5)2 – 27 (5) + 10


= 3 × 125 – 250 – 135 + 10


= 385 – 385


= 0


p (-2) = 3 (-2)3 – 10 (-2)2 – 27 (-2) + 10


= - 24 – 40 + 54 + 10


= - 64 + 64


= 0


And, p () = 3 ()3 – 10 ()2 – 21 () + 10




=


=


= 0


Verification of the relation is as follows:


Let us assume α = 5, β = - 2 and γ = 1/3


α + β + γ = 5 – 2 + 1/3 =



Also, + + = 5 (-2) + (-2) () + () (5)


= - 27/3


= - 9



And, αβγ =



Hence, verified


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