Choose the correct answer from the given four options in the following questions:
The number of polynomials having zeroes as $ -2 $ and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
Given:
Zeroes $-2$ and $5$.
To do:
We have to find the number of polynomials having zeroes as $-2$ and $5$.
Solution:
Let $\alpha=-2$ and $\beta=5$
Therefore,
$\alpha+\beta=-2+5=3$......(i)
$\alpha\beta=-2\times5=-10$ .......(ii)
This implies,
The polynomial $f( x)=x^2-( \alpha+\beta)x+\alpha\beta$
$\Rightarrow f( x)=x^2-3x-10$
If we multiply $f( x)$ by any constant $k$, roots will be $\alpha$ and $\beta$.
$\Rightarrow P( x)=k( x^2-3x-10)$
Thus, there are infinite number of polynomials possible.
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