Algebra

Polynomial Inequalities

Polynomial Inequalities: Level 3 Challenges

         

(n22)(n220)<0 \big(n^2-2\big)\big(n^2-20\big)<0

How many integers nn satisfy the inequality above?

Over the ranges

1v12u0.52z0.5 \begin{aligned} -1 \leq & v & \leq 1 \\ -2 \leq & u & \leq -0.5 \\ -2 \leq & z & \leq -0.5 \\ \end{aligned}

what is the range of w=vzuw = \frac {vz}{u} ?

If xx is an integer that satisfies the inequality

9<x2<99, 9 < x ^2 < 99,

find the difference between the maximum and minimum possible values of x.x.

As xx and yy ranges over all real values, what is the minimum value of

(15x+30y+20)2+(20x+40y+15)2? (15x + 30y + 20)^2 + (20x + 40y + 15)^2?

Given y=2x1+x2y=\frac{2x}{1+x^{2}}, where xx and yy are real numbers, what is the range of y2+y2y^{2}+y-2?

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