If $a$ and $b$ are the zeroes of $x^{2} - 2 x + 3$ , find a polynomial whose zeros are
$2 a + 3$ and $2 b + 3$

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Solution

Given that
$a$ and $b$ are the zeros of
$x^{2} - 2 x + 3$
Then,
We know that
Sum of zeros $= - \frac{c o e f f . o f x}{c o e f f . o f x^{2}}$
$a + b = - \frac{- 2}{+ 1}$
$a + b = 2 - - - - - - - - (1)$
Now,
Product of zeros $= \frac{c o n s t a n t t e r m}{c o e f f . o f x^{2}}$
$a . b = \frac{3}{1}$
$a b = 3 - - - - - - - - - - - (2)$
If $2 a + 3$ and $2 b + 3$ are the zeros of other polynomial.
Then
Sum of zeros $= 2 a + 3 + 2 b + 3$
$= 2 (a + b) + 6$
$= 2 (2) + 6$
$= 10$
Sum of zeros $= 10$
Product of zeros $= (2 a + 3) (2 b + 3)$
$= 4 a b + 6 a + 6 b + 9$
$= 4 a b + 6 (a + b) + 9$
$= 4 \times 3 + 6 \times 2 + 9$
$= 12 + 12 + 9$
$= 24 + 9$
Product of zeros $= 33$
Now,
Equation of polynomial
$x^{2} -$ (Sum of zeros) $x +$ product of zeros $= 0$
$x^{2} - 10 x + 33 = 0$
Hence, this is the answer.

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