Question:
Find a quadratic polynomial with zeroes $3+\sqrt{2}$ and $3-\sqrt{2}$
Solution:
Given that the zeroes of the quadratic polynomial are $3+\sqrt{2}$ and $3-\sqrt{2}$.
We have to find the quadratic polynomial from the given zeroes.
Let we assume that,
$\alpha=3+\sqrt{2}$
$\beta=3-\sqrt{2}$, then
$\alpha+\beta=3+\sqrt{2}+3-\sqrt{2}$
$=6$
$\alpha \beta=(3+\sqrt{2})(3-\sqrt{2})$
$=7$
Therefore the quadratic equation is given by
$x^{2}-(\alpha+\beta) x+\alpha \beta=x^{2}-6 x+7$
Hence the desire polynomial is $x^{2}-6 x+7$
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