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

It is given in the question that,

p (x) = x^{3} – 2x^{2} – 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