Question

Find an appropriate function $f\left(x\right)$ whose nature is shown in the graph above. $(q,r,s$ and $t$ are distinct real numbers$)$.

• A. $f\left(x\right)={(x-q)}^2$
• B. $f\left(x\right)=(x-r)(x+s)$
• C. $f\left(x\right)=(x-r)(x+s)(x+t)$
• D. $f\left(x\right)=(x-q)(x-r)(x+s)(x+t)$

Answer 1

The correct option is C $f\left(x\right)=(x-r)(x+s)(x+t)$

The number of zeroes of the polynomial $p\left(x\right)=$ number of times graph of $p\left(x\right)$ intersect the $x-axis$.

Given that polynomial $f\left(x\right)$ as shown in the figure intersects $x$ - axis $3$ times.

$\Rightarrow$ polynomial $f\left(x\right)$ must have $3$ Real roots.

Out of $3$ roots, two are negative and one positive.

Now Looking at Option $C$

$f\left(x\right)=(x-r)(x+s)(x+t)$

It has three roots i.e $r,-s,-t$

Hence, option $C$ is the correct answer