Question

nd the quadratic polynomial whose zeros are 2 and –6. Verify the relationship between the coefficients and the zeros of the polynomial.

Answer 1

Given: the quadratic polynomial whose zeros are 2 and –6.

To find: Verify the relationship between the coefficients and the zeros of the polynomial.

Solution:

• Now we know that sum of zeros is = 2+ (-6) = 2 - 6 = -4

• and product of zeros is 2 x -6 = -12

• The quadratic equation formula is:

                   x^2 - (a+b)x + ab = 0

• The quadratic equation will be:

                   x^2 - ( -4 )x + ( -12 ) = 0

                   x^2 + 4x - 12 = 0

• Now, the relation between the coefficient and the zeros of the polynomial is sum of zeroes and product of zeroes.

                   Sum of zeros is -b/a = -4 / 1 = -4

                   Product of zeros = c/a = -12 / 1 = -12

Answer:

              So the quadratic equation is x^2 + 4x - 12 = 0. And the results are verified in the solution.

Answer 2

GIVEN :

For the quadratic polynomial whose zeros are 2 and -6.

TO FIND :

The  quadratic polynomial and verify the relationship between the coefficients and the zeros of the polynomial.

SOLUTION :

Given that the zeros  of the quadratic equation are 2 and -6

Let $\alpha=2$ and $\beta=-6$ be the roots of the quadratic equation.

$sum of the roots=\alpha+\beta$

$=2-6$

=-4

∴ sum of the roots=-4

$Product of the roots=\alpha\times \beta$

$=2(-6)$

=-12

∴ Product of the roots=-12

With the given roots we can write the quadratic equation

$x^2-(sum of the roots)x+product of the roots=0$

Substitute the values we get,

$x^2-(-4)x+(-12)=0$

$x^2+4x-12=0$

∴ the quadratic equation is $x^2+4x-12=0$

Now verify with the coefficients and the zeros of the polynomial:

Here a=1 , b=4 and c=-12

$sum of the roots=\frac{-b}{a}$

$=\frac{-4}{1}$

$=-4$

∴ sum of the roots=-4

$product of the roots=\frac{c}{a}$

$=\frac{-12}{1}$

$=-12$

∴ product of the roots=-12

Hence verified.