Verify that 5, -2 and 1/3 are the zeroes of the cubic polynomial p(x)

1.	Verify that 5, -2 and 1/3 are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeroes and coefficients.
Answer» Let f(x) = 3x³ – 10x² – 27x + 10 5, -2 and 1/3 are the zeros of the polynomial (given) Therefore, f(5) = 3(5)³ – 10(5)² – 27(5) + 10 = 3 × 125 – 250 – 135 + 10 = 0 f(-2) = 3(-2)³ – 10(-2)² – 27(-2) + 10 = – 24 – 40 + 54 + 10 = 0 f(1/3) = 3 (1/3)³ – 10(1/3)² – 27(1/3) + 10 = 1/8 – 10/9 – 9 + 10 = 0 Verify relations: General form of a cubic equation: ax³ + bx² + cx + d. Now, Consider α = 5, β = – 2 and γ = 1/3 α + β + γ = 5 – 2 + 1/3 = 10/3 = -b/a αβ + βγ + αγ = 5 (-2) + (-2) (1/3) + (1/3) (5) = -9 = c/a And, αβγ = 5 (-2) (1/3)= -10/3 = -d/a

Verify that 5, -2 and 1/3 are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeroes and coefficients.

Answer»

Let f(x) = 3x³ – 10x² – 27x + 10

5, -2 and 1/3 are the zeros of the polynomial (given)

Therefore,

f(5) = 3(5)³ – 10(5)² – 27(5) + 10

= 3 × 125 – 250 – 135 + 10

= 0

f(-2) = 3(-2)³ – 10(-2)² – 27(-2) + 10

= – 24 – 40 + 54 + 10

= 0

f(1/3) = 3 (1/3)³ – 10(1/3)² – 27(1/3) + 10

= 1/8 – 10/9 – 9 + 10

= 0

Verify relations:

General form of a cubic equation: ax³ + bx² + cx + d.

Now,

Consider α = 5, β = – 2 and γ = 1/3

α + β + γ = 5 – 2 + 1/3 = 10/3 = -b/a

αβ + βγ + αγ = 5 (-2) + (-2) (1/3) + (1/3) (5) = -9 = c/a

And, αβγ = 5 (-2) (1/3)= -10/3 = -d/a