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The sum of all possible values of p for which atleast one root of equation x 4 – 8 x 3 + 22 x 2 – 24 x + p = 0 is of multiplicity 2, is

The sum of all possible values of p for which atleast one root of equation x4 – 8x3 + 22x2 – 24x + p = 0 is of multiplicity 2, is

Grade:12

2 Answers

Vikas TU
14149 Points
4 years ago
Dear student Roots are muliple of 2 
Put 2 in the equation , 
16 – 64 + 88 – 96 + p = 0 
p = 56 
Put x = 4 
p = 0 
Put x =8 
p = – 1216 
Dear student , There is no patern observed , Please attach image of the question , 
We will help you 
Good Luck 
Aditya Gupta
2081 Points
4 years ago
once again vikas ans is wrong as he doesnt even understand the meaning of multiplicity. 
note that for a root q to have multiplicity of 2, (x – q)^2 should be a factor of the given polynomial.
let p(x)= x4 – 8x3 + 22x2 – 24x + p = 0
then x4 – 8x3 + 22x2 – 24x + p = (x – q)^2*g(x)
differentiate both sides
p’(x)= 4x^3 – 24x^2 + 44x – 24= 2(x – q)*g(x) + (x – q)^2*g’(x)= (x – q)*f(x) where f(x)= 2g(x) + (x – q)*g’(x) is a polynomial.
so, clearly x= q is a root of p’(x).
now, the roots of 4x^3 – 24x^2 + 44x – 24 are x= 1, 2, 3. so, q= 1, 2 or 3 should also be the root of p(x).
when q=1, p(1)= – 9+p= 0 or p= 9
when q=2, p(2)= – 8 + p= 0 or p= 8
when q=3, p(3)= – 9+p= 0 or p= 9
so, p= 8, 9
so, sum of values of p is 8+9
= 17
kindly approve :)
 

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