Question
Find the quadratic polynomial whose zeroes are $\sqrt{3}+\sqrt{5}$ and $\sqrt{5}-\sqrt{3}$.
✓
Answer
Quadratic polynomial in terms of $x$, where coefficient of $x$ is sum of zeros and constant term will be product of zeros.
$\Rightarrow x^2-(\sqrt{3}+\sqrt{5}+\sqrt{5}-\sqrt{3}) x+(\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{3})$
$\Rightarrow x^2-(2 \sqrt{5}) x+(5-3) $
$\Rightarrow x^2-2 \sqrt{5} x+2$
Hence the required quadratic polynomial is
$\Rightarrow x^2-2 \sqrt{5} x+2$
$\Rightarrow x^2-(\sqrt{3}+\sqrt{5}+\sqrt{5}-\sqrt{3}) x+(\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{3})$
$\Rightarrow x^2-(2 \sqrt{5}) x+(5-3) $
$\Rightarrow x^2-2 \sqrt{5} x+2$
Hence the required quadratic polynomial is
$\Rightarrow x^2-2 \sqrt{5} x+2$
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