Question

5. Alpha, beta, gemma are zeroes of polynomial x^3+px^2+qx+2 such that alphabeta+1=0. Find the value of 2p+q+5.

Standard:- 10

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Answer 1

Hi dear...
here is your answer....

Answer 2

Note: Here, I am writing Alpha as a, Beta as b, Gamma as c.

Given Equation is f(x) = x^3 + px^2 + qx + 2.

It is in the form of ax^2 + bx + c + d = 0, we get

a = 1, b = p, c = q, d = 2

We know that sum of zeroes = -b/a

  a + b + c = -p/1

a + b + c = -p  ------- (1)



We know that product of zeroes = c/a

ab + bc + ca = q/1

ab + bc + ca = q   --------- (2)


And 

abc = -d/a

abc = -2/1

abc = -2.   ------ (3)


Given Equation is ab + 1 = 0

= > ab = -1  ------- (4)


Substitute (4) in (3), we get

(-1)c = -2

c = 2  ------ (5)


We know that -p = a + b + c from (1),

We know that q = ab + bc + ca = q from (2).

Given Equation is 2p + q + 5:

= > -2(a + b + c) + (ab + bc + ca) + 5

= > -2(a + b + 2) + (-1 + 2b + 2a) + 5

= > -2a - 2b - 4 - 1 + 2b + 2a + 5

= > -5 + 5

= > 0.



Therefore the value of 2p + q + 5 = 0.


Hope this helps!