Question

if alpha and beta are the zeros of x square -2x-8 then form a polynomial whose zeros are 2/alpha and 2/beta

Answer 1

Answer:

Given:

• alpha and beta are the zeros of x square -2x-8

To find:

•  a polynomial whose zeros are 2/alpha and 2/beta

Pre-requisite Knowledge:

If α and β are zeros of the polynomial ,then,

• α + β = -b/a

• α * β = c/a

Solving Question:

We are given the polynomial and are asked to find the polynomial with the respective zeros  2/alpha and 2/beta

 For this case , we can find the zeros of the polynomial and then find the zeros of the second polynomial,then we can find the answer.

Solution:

x² -2x -8 =0

a + b = -2

a * b = -8

a = -4x and b = 2x

x² -2x -8 =0

⇒ x² - 4x + 2x -8 = 0

or, x(x -4) + 2(x - 4) = 0

or, (x-4) (x+2) = 0

or, x - 4 =0

⇒ x = 4

and

x +2 = 0

or, x = -2

α = 4 , β = -2

Then the zeros of the next polynomial

zeros are 2/alpha and 2/beta

2/alpha  = 2 /4 = 1/2

2/beta = 2/-2 = -1

Then, to find polynomial

α + β = -b/a

α * β = c/a

⇒ α + β = -b/a

or, 1/2 + -1 = $\dfrac{1-2}{2}=\dfrac{-1}{2}$

or, -1/2 = -b/a

⇒ b = 1 , a = 2

α * β = c/a

1/2 * (-1) = -1/2 = c/a

⇒ c = -1

∴ The polynomial is k(2x² + x - 1 ), where 'k' is a constant.