Question

If α and β are the zeroes of a quadratic polynomial such that α β + = 0 and α β − = 8. Find the quadratic polynomial having α and β as its zeroes.
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Answer 1

The quadratic polynomial is equal to $x^{2}$ - 8.

Step-by-step explanation:

Given,

α + β = 0 and αβ = - 8

To find, the quadratic polynomial = ?

If α and β be the roots of quadratic polynomial.

The quadratic polynomial:

$x^{2}$ - (α + β)x + αβ

= $x^{2}$ - (0)x + (- 8)

= $x^{2}$ - (0)x - 8

= $x^{2}$ - 8

∴ The quadratic polynomial = $x^{2}$ - 8

Thus, the quadratic polynomial is equal to $x^{2}$ - 8.

Answer 2

Therefore the quadratic polynomial is f(x) =k(x²-16)      [where k is a constant]

Step-by-step explanation:

Polynomial: A polynomial makes with variables and coefficients.

Quadratic Polynomial: A quadratic polynomial is function of one variable and the degree of the variable is 2.

Given that,

α+ β= 0......(1)

α- β= 8.....(2)

____________

2α=8

⇒α=4

Then putting α=4 in equation (1)

4+β=0

⇒β= -4

If a and b are two zeros of quadratic polynomial, then the quadratic polynomial is f(x)=(x-a)(x-b)

Therefore the quadratic polynomial is

f(x)=k(x-4)[x-(-4)]   [where k is a constant]

=k(x-4)(x+4)

=k(x²-4²)

=k(x²-16)