# Polynomials

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## Finding the Roots

If x4 – 8x3 + ax2 – bx + 16 = 0 has positive real roots, find a – b.
1. -8
2. 6
3. -12
4. -14

Choice A. -8

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## Detailed Solution

Let p, q, r, s be the roots of the equation
p + q + r + s = 8
pqrs = 16
This happens only when p = q = r = s = 2
${\frac{(p+q+r+s)}{4} = 2}$.
$\sqrt{pqrs} = 2$.
Arithemtic mean is equal to Geometric Mean. This is possible only when all the numbers are equal. (Note that the question says all roots are positive and real)
p = q = r = s = 2
pq + pr + ps + qr + qs + rs = a
24 = a
pqr + pqs + prs+ qrs = b
32 = b
So, a – b = 24 – 32 = – 8

Correct Answer: – 8.

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