Question

If α and β be the zeroes of the polynomial x² + 10x + 30, then find the quadratic polynomial whose zeroes are α + 2β and 2α + β.

Answer 1

Given : p(x) = x² + 10x + 30

Now,

Let the zeroes of the quadratic polynomial be

α^' = α + 2β , β' = 2α + β

Then, α’ + β’ = α + 2β + 2α + β = 3α + 3β = 3(α + β)

α'β’ = (α + 2β) ×(2α + β) = 2α² + 2β² + 5αβ

Sum of zeroes = 3(α + β)

Product of zeroes = 2α² + 2β² + 5αβ

Then, the quadratic polynomial

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

= x² – (3(α + β))x + 2α² + 2β² + 5αβ

= x² – 3( – 10)x + 2 (α² + β²) + 5(30) {from eqⁿ (1) & (2)}

= x² + 30x + 2(α² + β² + 2αβ – 2αβ) + 150

= x² + 30x + 2 (α + β)² – 4αβ + 150

= x² + 30x + 2( – 10)² – 4(30) + 150

= x² + 30x + 200 – 120 + 150

= x² + 30x + 230

So, the required quadratic polynomial is x² + 30x + 230