Polynomial Inequalities

Definition A polynomial inequality in one variable is an inequality that can be written in one of the forms

where $p$ is a polynomial with real coefficients. Its solution set is the set of all real numbers $x$ that make it true. An inequality between two polynomials can always be put in this form by subtracting one side from both sides, which does not change the solution set. Multiplying or dividing both sides by a positive constant keeps the direction of the inequality, and doing so by a negative constant reverses it. Multiplying or dividing by an expression that contains $x$ is safe only when that expression is positive for every real $x$, as $x^2 + 1$ is, or negative for every real $x$, in which case the inequality reverses. An expression such as $x$ is positive for some values, negative for others and zero at one, so dividing by it is not a valid step.

Definition The critical values of a polynomial inequality $p(x) > 0$, or of the same inequality with $\ge$, $<$ or $\le$, are the real zeros of $p$, that is, the real numbers $c$ with $p(c) = 0$. A polynomial is continuous on the whole real line, so it can change sign only at a critical value. On each stretch between two consecutive critical values, and to the left of the smallest one and to the right of the largest one, $p$ keeps a constant sign. If $p$ has no real zeros, it has one sign on the whole line. This meaning of critical value belongs to inequalities. Calculus uses the similar names critical point and critical number for something different, an interior point of the domain of a function where its derivative is zero or does not exist.

Definition Placing the critical values $c_1 < c_2 < \cdots < c_k$ on the real number line splits it into the $k + 1$ open intervals

Each critical value is marked with a closed dot when the inequality is $\ge$ or $\le$, because it makes $p(x) = 0$ and so satisfies the inequality. It is marked with an open dot when the inequality is strict, because then it does not.

Definition A test point for one of the open intervals is any single number chosen inside it. Since $p$ has constant sign on the interval, the sign of $p$ at the test point is the sign of $p$ at every point of that interval.

Definition Write $p$ as a nonzero constant times a product of factors, usually powers $(x - c)^m$ of linear factors together with any quadratic factors that have no real zeros. A sign chart lists the intervals of the number line in order and records, for each interval, the sign of every factor and then the sign of the product. A product of nonzero numbers is positive when it has an even number of negative factors and negative when it has an odd number, where a negative constant counts as a negative factor. A factor $(x - c)^m$ with $m$ a positive integer is negative exactly when $x < c$ and $m$ is odd. So the sign of $p$ changes at a critical value of odd multiplicity and stays the same at a critical value of even multiplicity.

Definition A quadratic $ax^2 + bx + c$ with real coefficients and discriminant $b^2 - 4ac < 0$ has no real zeros, and its value has the same sign as $a$ for every real $x$. Such a factor adds no critical values and does not affect where the product changes sign. Only its constant sign enters the sign chart.

Definition The solution set of $p(x) > 0$ is the union of the open intervals where $p$ is positive, and the solution set of $p(x) < 0$ is the union of those where $p$ is negative. For $p(x) \ge 0$ or $p(x) \le 0$, add every critical value to the matching union. A critical value whose neighboring intervals both have the wrong sign is still a solution of a non-strict inequality, and it appears in the solution set as a single point.