If $α$ and $β$ are the zeoes of the polynomial $2 x^{2} - 7 x + 5$ , then find a polynomial whose zeros are $2 α + 1$ and $2 β + 3$ .

x^{2} - 11 x + 24

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x^{2} - 10 x + 12

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x^{2} - 3 x + 4

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Solution

The correct option is A $x^{2} - 11 x + 24$
$2 x^{2} - 7 x + 5 = 0$
$⟹ 2 x^{2} - 5 x - 2 x + 5 = 0$
$⟹ x (2 x - 5) - 1 (2 x - 5) = 0$
$x = 1, x = \frac{5}{2}$
Let, $x = 1$ is $α$ and $x = \frac{5}{2}$ is $β$
$⟹ 2 α + 1 = 3$
And $⟹ 2 β + 3 = 8$
$∴ x^{2} - (3 + 8) x + (3 \times 8) = 0$
$∴ x^{2} - 11 x + 24 = 0$

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If α and β are the zeoes of the polynomial 2x2−7x+5, then find a polynomial whose zeros are 2α+1 and 2β+3.

The correct option is A x2−11x+242x2−7x+5=0⟹2x2−5x−2x+5=0⟹x(2x−5)−1(2x−5)=0x=1,x=52Let, x=1 is α and x=52 is β⟹2α+1=3And ⟹2β+3=8∴x2−(3+8)x+(3×8)=0∴x2−11x+24=0

If $α$ and $β$ are the zeoes of the polynomial $2 x^{2} - 7 x + 5$ , then find a polynomial whose zeros are $2 α + 1$ and $2 β + 3$ .