Find a quadratic polynomial whose zeroes are $\frac{3+\sqrt{5}}{5}$ and $\frac{3-\sqrt{5}}{5}$.

Given: Zeroes of the quadratic polynomial are $\frac{3+\sqrt{5}}{5}$ and $\frac{3-\sqrt{5}}{5}$.

To do: To write the polynomial with the given zeroes.

Solution:

As given, zeroes of the polynomial are $\frac{3+\sqrt{5}}{5}$ and $\frac{3-\sqrt{5}}{5}$.

Sum of the roots $=\alpha+\beta=\frac{3+\sqrt{5}}{5}+\frac{3-\sqrt{5}}{5}$

$=\frac{3+\sqrt{5}+3-\sqrt{5}}{5}$

$=\frac{6}{5}$

Product of the zeroes $\alpha\beta=( \frac{3+\sqrt{5}}{5})( \frac{3-\sqrt{5}}{5})$

$\Rightarrow \alpha\beta=\frac{( 3+\sqrt{5})( 3-\sqrt{5})}{5\times5}$

$\Rightarrow \alpha\beta=\frac{3^2-( \sqrt{5})^2}{25}$

$\Rightarrow \alpha\beta=\frac{9-5}{25}$

$\Rightarrow \alpha\beta=\frac{4}{25}$

Therefore, the polynomial: $x^2-( \alpha+\beta)x+( \alpha\beta)=0$

$\Rightarrow x^2-\frac{6}{5}x+\frac{4}{25}=0$

$\Rightarrow 25x^2-30x+4=0$

Thus, the polynomial is $25x^2-30x+4=0$.

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Updated on: 10-Oct-2022

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