Question

Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials.
$4x^2+5\sqrt{2}x-3$

Answer 1

$f\left(x\right)=4x^2+5\sqrt{2}x-3$
$=4x^2+6\sqrt{2}x-\sqrt{2}x-3$ [by splitting the middle term]
$=2\sqrt{2}x(\sqrt{2}x+3)-1(\sqrt{2}x+3)$
$=(\sqrt{2}x+3)(2\sqrt{2}x-1)$
So, the value of $4x^2+5\sqrt{2}x-3$ is zero when $\sqrt{2}x+3=0or2\sqrt{2}x-1=0$
i.e., When $x=-\frac{3}{\sqrt{2}}and\frac{1}{2\sqrt{2}}$
so, the zeroes of $4x^2+5x-3are-\frac{3}{\sqrt{2}}orx=\frac{1}{2\sqrt{2}}$
$\therefore$ sum of zeroes $=-\frac{3}{\sqrt{2}}+\frac{1}{2\sqrt{2}}$
$=-\frac{5}{2\sqrt{2}}=\frac{-5\sqrt{2}}{4}$
$=-\frac{\left(Coefficientofx\right)}{\left(Coefficientof{x}^2\right)}$
And product of zeroes $=(-\frac{3}{\sqrt{2}})\left(\frac{1}{2\sqrt{2}}\right)=-\frac{3}{4}$
$=\frac{constantterm}{coefficientof{x}^2}$
Hence, verified the relations between the zeroes and the coefficient of the polynomial