Question

\(\alpha, \beta\) are zeroes of the polynomial \(p(x) = 3x^2 - 6x - 5\). Find the value of \(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\).

Hint:

Avoid solving for \(\alpha\) and \(\beta\) individually using the quadratic formula unless the polynomial factors easily. Always aim to manipulate the expression into forms of \((\alpha+\beta)\) and \((\alpha\beta)\).

Answer 1

Step 1: Identify Coefficients
Given polynomial:
p(x) = 3x² − 6x − 5

So,
a = 3
b = −6
c = −5

Step 2: Find Sum and Product of Zeroes
Sum (α + β) = −b/a
= −(−6)/3
= 6/3
= 2

Product (αβ) = c/a
= −5/3

Step 3: Find α² + β²
Using identity:
α² + β² = (α + β)² − 2αβ

= 2² − 2(−5/3)
= 4 + 10/3
= (12 + 10)/3
= 22/3

Step 4: Evaluate Required Expression
1/α² + 1/β² = (α² + β²) / (αβ)²

= (22/3) / ( (−5/3)² )
= (22/3) / (25/9)

Multiply by reciprocal:
= (22/3) × (9/25)
= (22 × 3) / 25
= 66/25

Final Answer:
The required value is 66/25