Question

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

Answer 1

Given that

$a$ and $b$ are the zeros of

$x^2-2x+3$

Then,

We know that

Sum of zeros $=-\frac{coeff.ofx}{coeff.of{x}^2}$

$a+b=-\frac{-2}{+1}$

$a+b=2--------\left(1\right)$

Now,

Product of zeros $=\frac{constantterm}{coeff.of{x}^2}$

$a.b=\frac{3}{1}$

$ab=3-----------\left(2\right)$

If $2a+3$ and $2b+3$ are the zeros of other polynomial.

Then

Sum of zeros $=2a+3+2b+3$

$=2(a+b)+6$

$=2\left(2\right)+6$

$=10$

Sum of zeros $=10$

Product of zeros $=(2a+3)(2b+3)$

$=4ab+6a+6b+9$

$=4ab+6(a+b)+9$

$=4\times 3+6\times 2+9$

$=12+12+9$

$=24+9$

Product of zeros $=33$

Now,

Equation of polynomial

$x^2-$ (Sum of zeros) $x+$ product of zeros $=0$

$x^2-10x+33=0$

Hence, this is the answer.