# Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:(i) (ii) (i) P(x) = Now for zeroes, putting the given values in x.

P(1/2) = 2(1/2)3 + (1/2)2 - 5(1/2) + 2
= (1/4) + (1/4) - (5/2) + 2
= (1 + 1 - 10 + 8)/2
= 0/2 = 0

P(1) = P(-2) = Thus, 1/2, 1 and -2 are zeroes of given polynomial.

Comparing given polynomial with ax3 + bx2 + cx + d and Taking zeroes as α, β, and γ, we have Now, We know the relation between zeroes and the coefficient of a standard cubic polynomial as Substituting value, we have  Since, LHS = RHS (Relation Verified)    Since LHS = RHS, Relation verified.   Since LHS = RHS, Relation verified.

Thus, all three relationships between zeroes and the coefficient is verified.

(ii) p(x) = x3 – 4x2 + 5x – 2

Now for zeroes , put the given value in x.

P(2) = = P(1) = P(1) = Thus, 2, 1 , 1 are the zeroes of the given polynomial.

Now,

Comparing the given polynomial with ax3 + bx2 + cx + d, we get Now,  4 = 4   5 = 5

αβγ = 2 × 1 × 1 = 2

2 = 2

Thus, all three relationships between zeroes and the coefficient is verified.

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