Question

Question 1 (vii)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials

(vii) $2s^2-(1+2\sqrt{2})s+\sqrt{2}$

Answer 1

vii) Let $f\left(s\right)=2s^2-(1+2\sqrt{2})s+\sqrt{2}$
$=2s^2–s–2\sqrt{2}s+\sqrt{2}$
$=s(2s–1)–\sqrt{2}(2s–1)$
$=(2s-1)(s-\sqrt{2})$
$=(2s–1)(s-\sqrt{2})$
So, the value of $2s^2–(1+2\sqrt{2})s+\sqrt{2}$ is zero when $2s–1=0ors-\sqrt{2}=0$
i.e., when $s=\frac{1}{2}ors=\sqrt{2}$
So the zeroes of $2s^2–(1+2\sqrt{2})s+\sqrt{2}are\frac{1}{2}and\sqrt{2}$
$\therefore$ sum of zeroes = $\frac{1}{2}+\sqrt{2}=\frac{1+2\sqrt{2}}{2}=\frac{-[-(1+2\sqrt{2}\left)\right]}{2}\left(\frac{Coefficientofs}{Coefficientof{s}^2}\right)$
And product of zeroes =$\left(\frac{1}{2}\right)\left(\sqrt{2}\right)=\frac{1}{\sqrt{2}}=\frac{Constantterm}{Coefficientof{s}^2}$
Hence, verified the relations between he zeroes and the coefficients of the polynomial