Question

if alpha and beta are the zeroes of a polynomial 3x2+5x-2 then form a quadratic polynomial whose zeroes are3alpha+2beta and2alpha+3beta

Answer 1

Here is the stepwise solution. I hope u find it useful
Given:
p(x) = 3x² + 5x - 2
α & β are the zeroes
To Find:
A polynomial with zeroes 3α + 2β and 2α + 3β
In p(x)
α + β = -b/a
α + β = -5/3 

αβ = c/a
αβ = -2/3

Let the new polynomial g(x) = a'x² + b'x + c'
Zeroes of g(x) ⇒ 3α + 2β = α' & 2α + 3β = β'
α' + β' = -b'/a'
3α + 2β + 2α + 3β = -b'/a'
5(α + β) = 5(-5/3) = -b'/a'
-25/3 = -b'/a'

α'β' = c/a
(3α + 2β) x (2α + 3β) = c'/a'
6(α²+β²) +13αβ = c'/a'
6({α+β}² - 2αβ) + 13αβ = c'/a'
6(4/9 - 2(-2/3)) + 13(-2/3) = c'/a'
32/3 - 26/3 = c'/a'
6/3 = c'/a'

∴ The new polynomial is a'x² + b'x + c'
⇒3x² - 25x + 6

Answer 2

if alpha and beta are the zeroes of a polynomial 3x2+5x-2 then form a quadratic polynomial whose zeroes are3alpha+2beta and2alpha+3beta