Question

α, β, γ are zeroes of polynomial x³+ px²+qx+2 such that α.β + 1 = 0. Find the value of 2p + q + 5​

Answer 1

Answer:

Value of 2p + q + 5 = 0

Step-by-step explanation:

Step-by-step explanation:

If α, β and γ are three zeros of x³ + px² + qx + 2 then

(x - α)(x - β)(x - γ) = 0

(x² - αx - βx + αβ)(x - γ) = 0

x³ - αx² - βx² + αβx - γx² + αγx + βγx - αβγ = 0

x³ - x²(α + β + γ) + x(αβ + βγ + αγ) - αβγ = 0

Now we compare it with x³ + px² + qx + 2 = 0

Then  p = -(α + β + γ), q = (αβ + βγ + αγ) and αβγ = -2

Moreover this αβ + 1 = 0

hope it is helpful to you

Or αβ = -1

Since αβγ = -2 ⇒ (-1)γ = -2

⇒ γ = 2

Now p = -(α + β + 2),  q = (-1 + 2β + 2α)

Now we have to evaluate 2p + q + 5

-2(α + β + 2) + (-1 + 2β + 2α) + 5 = -2α - 2β - 4 - 1 + 2α + 2β + 5 = 0

Therefore, 2p + q + 5 = 0