Question

Find a quadratic polynomial whose zeroes are 1 and -3. Verify the relation between the coefficient and zeros of the polynomial.

Answer 1

The zeroes are given :-
1 and (-3)

Now
Sum of the zeroes is
$1 + ( - 3) = 1 - 3 = ( - 2) \: \: \: .....(1)$

Now,
Product of the zeroes is
$1 \times ( - 3) = ( - 3) \: \: .......(2)$

Now,

We know that the form of any quadratic polynomial is

$k( {x}^{2} - (sum \: of \: zeroes)x + product)$
So,

Putting the values in the form mentioned above :-

$k( {x}^{2} - ( - 2)x + ( - 3)) \\ \\ k( {x}^{2} + 2x - 3) \\ \\ when \: k = (1) \\ \\ polynomial \: becomes \\ \\ {x}^{2} + 2x - 3$


Now,

The sum of zeroes is - b/a

That is - 2/1 = (-2)
Same as in.... (1)


Product of the zeroes is c/a

That is - 3/1 = (-3)
Same as in..... (2)


Hence,
The polynomial is (x²+2x-3)


Also, The relation between the zeroes and the coefficient of the quadratic polynomial is verified.