Question

obtain a quadratic polynomial whose zeros are 2alpha+beta and alpha+2beta where alpha and beta are the zeros of x​2​​−3x−8

Answer 1

Hey, 
Here is Your Answer,
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First You Need to Find the sum of zeroes and product of zeroes of given Ploynomial:-      (x²+3x-10)

Sum of Zeroes = -b/a
α+β= -(3/1)
α+β= -3   .................................................(i)


Product of Zeroes = c/a
αβ = -10

Now You Find the Sum and Product of Zeroes by Given zeroes :-
1) 2α+β
2) α+2β

Sum of Zeroes
2α+β+α+2β = 3α+3β 
3(α+β)
3(-3)       ..........................[using (i)]
-9

Product of Zeroes
(2α+β)(α+2β)
2α(α+2β)+β(α+2β)
2α²+4αβ+αβ+2β²
2(α²+β²)+5αβ
2[(α+β)²-2αβ]+5αβ      .......................[ α²+β²=(α+β)²-2αβ]
2(α+β)²-4αβ+5αβ
2(α+β)²+αβ
2(-3)²-10      .................................................. [using (i) & (ii) ]
2(9)-10
18-10
8

Now Find the New Quadratic Polynomial
      x²-(sum of zeroes)x+(product of zeroes)
      x²-(-9)x+8
      x²+9x+8 is the Required polynomial 
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(I HOPE IT HELPS YOU)
MARK AS A BRILLIANT