Octopus Poly

Algebra Level 5

Let

$\displaystyle f(x) = 5x^{88} + ax^{5} + bx^{2} - cx -10$

Given

$a, b, c \in \mathbb R$

$f(2) = 5, f(4) = -5$

Then find the minimum number of real roots of $f(x)$

The answer is 4.

2 solutions

Dheeraj Agarwal
Jan 19, 2015

use descartes theorm

Omkar Kamat
Jan 3, 2015

A polynomial is dominated by the term with the largest power for very large and very small powers of x. Hence, for very small values of x, we have f(x)>0. f(0)=-10. So by continuity of polynomials, there must be a root in between. f(2)>0 so again by the same argument there must be a root here.

f(4)<0 so there must be another root. Now for very large x, f(x)>0 so we have another root. So the minimum number of real roots has to be 4.

Interesting thing is that for the any even degree(highest in polynomial) greater than 4, answer remains the same.

Krishna Sharma - 6 years, 5 months ago

Octopus Poly

The answer is 4.

2 solutions

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