Question

find a quadratic polynomial whose zeros are 1 and -3 verify the relation the coefficient and zeros of the polynomial

Answer 1

1 and -3 are the zeroes of a polynomial.
Let the polynomial be p(x) .
Let
$\alpha = 1$
and
$\beta = - 3$
So,
$\alpha \: and \: \beta$
are zeroes of the polynomial p(x).

We know,
$\alpha + \beta = - b \div a$
and ,
$\alpha \times \beta = c \div a$
So, let's solve the first formula ..that is ,
$\alpha + \beta = - b \div a$
We know that
$\alpha + \beta = 1 + ( - 3) \\ = 1 - 3 \\ = - 2 \\ also \\when \: \alpha + \beta = - 2 \\ - b \div a = - 2 \\ because \: \\ \alpha + \beta = - b \div a$
Now let's solve the second formula ,
$\alpha \times \beta = c \div a \\ \alpha \times \beta = 1 \times - 3 \\ = - 3 \\ so \: if \: \alpha \times \beta = - 3 \\ then \: \\ c \div a = - 3$
...if we assume that a = 1
then b= 2 and c = -3
therefore the polynomial p(x) will be
${x}^{2} + 2x + 3$