Question

If α and β are zeroes of a polynomial 6x² − 5x + 1 then form a quadratic polynomial whose zeroes are α² and β².

Answer 1

Let P(x) = 6x² − 5x + 1

The roots of P(x) are α and β.

So,

Sum of roots (α + β) = `(−b)/a`

= `(−(−5))/6`

= `5/6`

and,

Product of roots (αβ) = `c/a`

= `1/6`

Quadratic polynomial whose zeroes are α² and β².

So,

Required polynomial = q(x)

= x² − (sum of zeroes)x + product of zeroes

= x² − (α² + β²) x + α² β²

= x² − [(α + β)² − 2αβ] x + (αβ)²

= `x^2 − [(5/6)^2 − 2 × 1/6]x + (1/6)^2`

= `x^2 − (25/36 − 1/3) x + 1/36`

= `x^2 − ((25 − 12)/36) x + 1/36`

= `x^2 − 13/36 x + 1/36`

= 36x² − 13x + 1

⇒ Required polynomial is 36x² − 13x + 1.