# Finding a polynomial with product and sum of its zeroes

A was reading a book with this question in it:

Q. Find a quadratic polynomial, the sum of whose zeroes is 7 and their product is 12. Hence find the zeores of the polynomial.
Sol.
Let the polynomial be $ax^2+bx+c$ and suppose its zeroes are $\alpha$ and $\beta$
Therefore, sum of zeroes $=\alpha+\beta=\frac{-b}{a}=7$
and product of zeores $=\alpha\beta=\frac{c}{a}=12$
Take $a=LCM(12,1)=12$
Therefore $b=-7a=-7\times12=-84$ and $c=12a=12\times12=144$
So the polynomial is $12x^2-84x+144$

$\cdots$

Why have we taken $a=LCM(12,1)$? If we had taken $a=1$ then we could have got the answer without any kind of calculation. Is there any real reason for taking $a=LCM(12,1)$?

There is a reason for choosing $a = LCM (1, 1)$, as this gives us an integer polynomial whose coefficients are the smallest.