Question

if alpha and beta are zeroes of the polynomials x^2-2x-15 then form a quadratic polynomial whose zeroes are2α and2β

Answer 1

we know that , sum of zeroes=

$\alpha + \beta = \frac{ - b}{a} = \frac{2}{1} = 2$
___________________________(1)
also,

product of zeroes=

$\alpha \beta = \frac{c}{a} = \frac{ - 15}{1} = - 15$
___________________________(2)
now,

from (1)
$2 \alpha + 2 \beta = 2( \alpha + \beta ) \\ = 2(2) = 4$
from (2),

$2 \alpha 2 \beta = 4 (\alpha \beta ) \\ = 4( - 15) = - 60$
therefore the quadratic polynomial=

✓ x²-4x-60

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

Answer 2

Given Quadratic Equation is x^2 - 2x - 15.

Here, a = 1, b = -2, c = -15.

(i)

We know that Sum of zeroes = -b/a

⇒ α + β = -(-2)

⇒ α + β = 2.

(ii)

We know that product of zeroes = c/a

⇒ αβ = -15

Given that 2α and 2β are the zeroes of the quadratic polynomial.

Sum of zeroes:

⇒ 2α + 2β

⇒ 2(α + β)

⇒ 4

Product of zeroes:

⇒ 2α * 2β

⇒ 4(αβ)

⇒ 4(-15)

⇒ -60.

Therefore, the required quadratic polynomial is:

⇒ x^2 - (Sum of zeroes)x + Product of zeroes

⇒ x^2 - (4)x + (-60)

⇒ x^2 - 4x - 60.

Hope it helps!