Question

Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2, respectively.

Answer 1

Given,

The sum of zeros of a quadratic polynomial is -3.

The product of zeros of a quadratic polynomial is 2.

To find out,

The quadratic polynomial.

Solution:

$\displaystyle{if \: \alpha + \beta \ are \: the \: zeroes \: of \: the \: quadratic \: polynomial}\\\\\displaystyle{{ax}^{2} + bx + c \: where \: a \: is \: not \: equal \: to \:0 \: then \: \alpha +\beta = \dfrac{ - b}{a} and \: \alpha \beta=\dfrac{c}{a}}\\\\\display \text{As it is given}$$\displaystyle{\alpha +\beta = -3}\\\\\displaystyle{\alpha\times\beta =2}\\\\\display \text{the \: quadratic \: polynomial }\\\\\displaystyle{{ax}^{2} + bx + c \: is \: k( {x}^{2} - ( \alpha+\beta )x+\alpha \beta )}\\\\\display \text{where \: k \: is \: a \: constant \: and \: k \: is \: not \: equal \: to \: 0}$$\displaystyle{k( {x}^{2} - ( - 3)x + 2)}\\\\\displaysyle{k( {x}^{2} + 3x + 2)}\\\\\display \text{We can take different values for k}\\\\\display \text{therefore the quadratic polynomial is {x}^{2} + 3x + 2 }$

Answer 2

Answer:

Given that sum of the zeros of the polynomial is equals to -3

And

product of the zeros of the polynomial is 2.

As we know that

any polynomial whose sum of roots and product of roots are given will be of the form

${x}^{2} - sx + p$

Where s is the sum of the zeros of the polynomial and p is product of the zeros of the polynomial.

So polynomial will be:

${x}^{2} + 3x + 2$