Verify that 3, ‒2, 1 are the zeros of the cubic polynomial p(x) = x3 – 2x2 – 5x + 6 and verify the relation between its zeros and coefficients.

It is given in the question that,

p (x) = x3 – 2x2 – 5x + 6


Also, 3, -2 and 1 are the zeros of the given polynomial


∴ p (3) = (3)3 – 2 (3)2 – 5 (3) + 6


= 27 – 18 – 15 + 6


= 33 – 33


= 0


p (-2) = (-2)3 – 2 (-2)2 – 5 (-2) + 6


= -8 – 8 + 10 + 6


= - 16 + 16


= 0


And, p (1) = (1)3 – 2 (1)2 – 5 (1) + 6


= 1 – 2 – 5 + 6


= 7 – 7


= 0


Verification of the relation is as follows:


Let us assume = 3, = - 2 and = 1


= 3 – 2 + 1


= 2


∴


Also, + + = 3 (-2) + (-2) (1) + 1 (3)


= - 6 – 2 + 3


= - 5


∴


And, = 3 × (-2) × 1


= - 6


∴


1