Question

Zeroes of the quadratic polynomial x² − 3x + 2 are α and β. Construct a quadratic polynomial whose zeroes are 2α + 1 and 2β + 1.

Answer 1

Given that α and β are zeroes of the Quadratic Polynomial x² − 3x + 2,

∴ Sum of roots `(α + β) = (-"b")/"a"`

= `(-(-3))/1`

= 3

∴ Product of roots (αβ) = `"c"/"a"`

= `2/1`

= 2

Quadratic Polynomial whose zeroes are 2α + 1 and 2β + 1 is:

Sum of roots = 2α + 1 + 2β + 1

= 2α + 2β + 2

= 2(α + β + 1)

= 2(3 + 1)   ...[∵ α + β = 3]

= Product of roots = (2α + 1) (2β + 1)

= 4αβ + 2α + 2β + 1

= 4αβ + 2(α + β) + 1

= 4 × 2 + 2(3) + 1

= 8 + 6 + 1

= 15

(∵ αβ = 2 and α + β = 3 from above)

∴ Quadratic Polynomial ⇒ x² − (Sum of the roots x + Product) = 0

⇒ x² − 8x + 15

= 0