Question

Find all the zeroes of polynomial 2x4-11x3+7x2+13x-7 it being given that two of it zeroes are 3+✓2 and 3-√2

Answer 1

$\text{Given polynomial is } 2x^4-11x^3+7x^2+13x-7$

$\text{Sum of the zeros =}3+\sqrt{2}+3-\sqrt{2}=6$

$\text{Product of the zeros }$

$=(3+\sqrt{2})(3-\sqrt{2})$

$=3^2-(\sqrt{2})^2$

$=9-2=7$

$\therefore\text{Corresponding factor is }\bf\,x^2-6x+7$

$\text{Now,}$

$2x^4-11x^3+7x^2+13x-7=(x^2-6x+7)(2x^2+px-1)$

$\text{Equating coefficients of x on both sides we get }$

$13=6+7p$

$7p=7$

$\implies\,p=1$

$\therefore\text{Other factor is }\,2x^2+x-1$

$2x^2+x-1=(2x-1)(x+1)$

$\therefore\text{All zeros are }-1,\;\frac{1}{2},\;3+\sqrt{2},\;3-\sqrt{2}$