Polynomial Inequalities

A polynomial inequality is simply a question like "when is $x^2 - x - 6$ positive?" The formula up top is really a recipe: $p(x) > 0$ (the question, with zero on one side), the zeros of $p$ (the only places it can switch from positive to negative) and testing each interval (checking one number in each stretch between them). That's the whole method. It's quicker than it sounds.

Why it works. Picture the graph of $p(x)$. To go from above the x-axis to below it, the smooth curve has to cross the axis, and crossing the axis means $p(x) = 0$. So between two neighboring zeros, the sign can't change, and checking any single point tells you the sign for the whole stretch. This is why we want zero on one side: we're asking about the sign of one expression, not comparing two.

Steps:

1. Rewrite as $p(x) > 0$ (or $<$, $\ge$, $\le$) with zero on one side.
2. Factor and find the zeros.
3. Make a sign chart: test one point in each interval.
4. Pick the intervals with the right sign; include the zeros only for $\ge$ or $\le$.

A common mistake. In something like $x^2 > 3x$, it's tempting to divide both sides by $x$ and get $x > 3$. But you don't know whether $x$ is positive or negative, and dividing by a negative would flip the sign. That shortcut loses every negative solution. Move everything over instead: $x^2 - 3x > 0$, so $x(x - 3) > 0$, which is true for $x > 3$ and also for $x < 0$.

Engineers find the safe operating ranges where a stress polynomial stays below a limit; economists find price ranges where profit is positive.