Question

If a1, a2, and a3 are the zeroes of polynomial
p(x) = 3x^3 + 4x^2 + 2x + 4, then the value of
a1^2 + a2^2 + a3^2 is equal to​

Answer 1

Answer:

a₁² + a₂² + a₃² = 4/9

Step-by-step explanation:

The given polynomial is

p(x) = 3x³ + 4x² + 2x + 4

Since a₁, a₂, a₃ are the zeroes of p(x), then by the relation between zeroes and coefficients, we get

a₁ + a₂ + a₃ = - 4/3 ..... (1)

a₁a₂ + a₂a₃ + a₃a₁ = 2/3 ..... (2)

a₁a₂a₃ = - 4/3 ..... (3)

Using the identity formula

(a + b + c)² = a² + b² + c² + 2 (ab + bc + ca), we get

a₁² + a₂² + a₃²

= (a₁ + a₂ + a₃)² - 2 (a₁a₂ + a₂a₃ + a₃a₁)

= (- 4/3)² - 2 (2/3), by (1) and (2)

= 16/9 - 4/3

= (16 - 12)/9

= 4/9