Question

if alpha and beta are zeroes if the polynomial 2xsquare -5x+7,find a polynomial whose zeroes are 2alpha+3beta and 3alpha +2beta

Answer 1

polynomial is ×^2+25/×+ 137/2


or 2×^2 + 25×. + 137

hope this helps you
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Answer 2

If α and β are zeroes of polynomial 2x² - 5x + 7 then,

Sum of zeros = - Coefficient of x/Coefficient of x²

⇒ α + β = - (-5)/2

⇒ α + β = 5/2

Product of zeros = Constant Term/Coefficient of x²

⇒ αβ = 7/2

Therefore, (α + β)² = α² + β² + 2αβ

⇒ 5²/2² = α² + β² + 2(7/2)

⇒ 25/4 - 7 = - 3/4 = α² + β²

S' of zeros = 2α + 3β + 3α + 2β

⇒ 5α + 5β = 5(α + β)

⇒ 5(5/2) = 25/2

P' of zeros = (2α + 3β)(3α + 2β)

⇒ 6(α² + β²) + 9αβ + 4αβ

⇒ - 18/4 + 63/2 + 28/2

⇒ ( -9 + 63 + 28 )/2 = 41

Polynomial = x² - Sum of zeros + Product of zeros

⇒ x² - 25x/2 + 41

Answer : x² - 25x/2 + 41