CAT Quantitative Aptitude Questions | CAT Algebra - Polynomials questions

The question is about Finding the Roots of a bi-quadratic equation. Sum and product of the roots of the polynomial is given and we need to find out an algebraic expression of roots. Polynomials is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. Clue: This question involves a particular rule from inequality.

Question 6: If x⁴ – 8x³ + ax² – bx + 16 = 0 has positive real roots, find a – b.

1. -8
2. 6
3. -12
4. -14

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Polynomials: What would happen to AM and GM when the roots are equal?

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.
√(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

The question is "If x⁴ – 8x³ + ax² – bx + 16 = 0 has positive real roots, find a – b."

Hence the answer is "-8"

Choice A is the correct answer.