Question

Find the zeros of polynomial $5x^2-3$ and Verify the relationship between zeros and co-efficient

Answer 1

We first find the zeroes of the polynomial $5x^2-3$ by equating it to $0$ as shown below:

$5x^2-3=0\Rightarrow 5x^2=3\Rightarrow x^2=\frac{3}{5}\Rightarrow x=\pm \sqrt{\frac{3}{5}}\Rightarrow x=\sqrt{\frac{3}{5}},x=-\sqrt{\frac{3}{5}}$

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

Now, consider the sum of the zeroes as follows:

$\sqrt{\frac{3}{5}}+(-\sqrt{\frac{3}{5}})=\sqrt{\frac{3}{5}}-\sqrt{\frac{3}{5}}=0=\frac{0}{5}=-\left(\frac{coefficientofx}{coefficientof{x}^2}\right)$

And the product of the zeroes is:

$\sqrt{\frac{3}{5}}\times (-\sqrt{\frac{3}{5}})=-\sqrt{\frac{3\times 3}{5\times 5}}=-\frac{3}{5}=\frac{constantterm}{coefficientof{x}^2}$

Hence, the relationship between zeroes and coefficient is verified.