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
∴