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

It is given in the question that,

p (x) = 3x^{3} – 10x^{2} – 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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