Question

Find the quadratic polynomial whose zeros are 2 and -6.Verify the relation between the coefficient and the zeros of the polynomial ​

Answer 1

GIVEN:

Zeroes of polynomial are 2 and -6

TO FIND:

The quadratic polynomial and verify relationship between coefficientand the zeroes

SOLUTION:

We know that,

The standard form of quadratic polynomial is,

x² - (sum of zeroes)x + product of zeroes

Sum of zeroes = α + β = 2 + (-6) = -4

Product of zeroes = αβ = 2 × -6 = -12

=> x² - (α+β)x - (αβ)

=> x² - (-4)x + (-12)

=> x² + 4x - 12

Thus, x² + 4x - 12 is the required polynomial.

VERIFICATION:

Sum of zeroes = α + β = -b/a = -4/1 = -4

=> α + β = -4

=> -4 = -4

Product of zeroes = αβ = c/a = -12/1 = -12

=> αβ = -12

=> -12 = -12

Hence, verified!

Answer 2

$\huge\mathfrak\blue{Answer:}$

Given:

We have been given the two zeroes of a quadratic polynomial, ie 2 and -6.

To Find:

We need to find the quadratic polynomial and also Verify the relation between the coefficient and the zeros of the polynomial.

Solution:

Let 2 be α and -6 be β.

Sum of zeroes α + β = 2 + (-6) = 2 - 6 = -4

Product of zeroes αβ = 2 × (-6) = -12

We know that the the form of any quadratic polynomial is:

$k( {x}^{2} - ( \alpha + \beta )x + \alpha \beta )$

$k( {x} {}^{2} - ( - 4x) + ( - 12))$

$k( {x}^{2} + 4x - 12)$

Hence the required polynomial is

(x^2 + 4x - 12)

Now, inorder to verify the relation between the coefficient and the zeros of the polynomial we have,

Sum of zeroes = -b/a = -4/1 = -4.

Product of zeroes = c/a = -12/1 = -12.

Hence verified!