Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
Question:

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

$3 x^{2}-x-4$

Solution:

Let $f(x)=3 x^{2}-x-4$

To find the zeroes, we put $f(x)=0$

$\Rightarrow 3 x^{2}-x-4=0$

$\Rightarrow 3 x^{2}-4 x+3 x-4=0$

$\Rightarrow x(3 x-4)+1(3 x-4)=0$

$\Rightarrow(x+1)(3 x-4)=0$

$\Rightarrow(x+1)=0$ or $(3 x-4)=0$

$\Rightarrow x=-1, \frac{4}{3}$

Hence, all the zeroes of the polynomial $f(x)$ are $-1$ and $\frac{4}{3}$.

Now,

$f(-1)=3(-1)^{2}-(-1)-4$

$=3(1)+1-4$

$=3-3$

$=0$

$f\left(\frac{4}{3}\right)=3\left(\frac{4}{3}\right)^{2}-\frac{4}{3}-4$

$=3\left(\frac{16}{9}\right)-\frac{4}{3}-4$

$=\frac{16}{3}-\frac{4}{3}-4$

$=\frac{12}{3}-4$

$=4-4$

$=0$

Hence, the relationship between the zeros and the coefficients is verified.