Question

Find all zeros of the polynomial 2x⁴ – 9x³ + 5x² +3x – 1, if two of its zeros are $2+\sqrt{3}\mathrm{and}2-\sqrt{3}$.

Answer 1

It is given that $2+\sqrt{3}$ and $2-\sqrt{3}$ are two zeroes of the polynomial f(x) = 2x⁴ – 9x³ + 5x² + 3x – 1.

$\begin{array}{rcl}\therefore \left\{x-\left(2+\sqrt{3}\right)\right\}\left\{x-\left(2-\sqrt{3}\right)\right\}& =& \left(x-2-\sqrt{3}\right)\left(x-2+\sqrt{3}\right)\\ & =& {\left(x-2\right)}^2-{\left(\sqrt{3}\right)}^2\\ & =& x^2-4x+4-3\\ & =& x^2-4x+1\end{array}$

is a factor of f(x)
Now, divide f(x) by x² – 4x + 1.




∴ f(x) = (x² – 4x + 1) (2x² – x – 1)
Hence, other two zeroes of f(x) are the zeroes of the polynomial 2x² – x – 1.
2x² – x – 1 = 2x² – 2x + x – 1 = 2x(x – 1) + 1 (x – 1) = (2x + 1) (x – 1)
Hence, the other two zeroes are $-\frac{1}{2}$ and 1.