If $α$ and $β$ are the zeroes of the quadratic polynomial $f (x) = 2 x^{2} - 5 x + 7$ find a polynomial where zeroes are $2 α + 3 β$ and $3 α + 2 β$

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Solution

When roots are given, equation in quadratic form
$x^{2} - (α + β) x + α β = 0$
Here roots are $(2 α + 3 β)$ and $(3 α + 2 β)$
$∴ x^{2} - (2 α + 3 β + 3 α + 2 β) x + (2 α + 3 β) (3 α + 2 β) = 0$
$x^{2} - 5 (α + β) x + 6 α^{2} + 4 α β + 4 α β + 6 β^{2} = 0$
$f (x) = 2 x^{2} - 5 x + 7$
$α + β = \frac{5}{2}$ , $α β = \frac{7}{2}$
$∴ α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= \frac{25}{4} - 7 = \frac{- 3}{4}$
$x^{2} - 5 (\frac{5}{2}) x + 6 (- \frac{3}{4}) + 13 \times \frac{7}{2} = 0$
$∴ 2 x^{2} - 50 x - 18 + 182 = 0$
$2 x^{2} - 50 x + 164 = 0$
$∴ x^{2} - 25 x + 82 = 0$ .

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If $α a n d β$ are the zeroes of the polynomial $a x^{2} + b x + c$ , then find the polynomial whose zeroes are $2 α + 3 β, 3 α + 2 β$ .

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