Polynomial Inequalities

Definition

A polynomial inequality is an expression that compares a polynomial function to zero (e.g., $f(x) > 0$). Solving it involves finding the intervals of $x$ for which the inequality is true.

• How to read: “The polynomial f of x is strictly greater than zero.”
• Meaning: Find all $x$ where the polynomial expression is positive, negative, or zero using sign analysis on intervals.

Why It Matters

Most important questions are inequalities: “Is profit $> 0$?”, “Is stress $<$ failure point?”. These equations help us find the exact “Intervals of Safety.” It is the math of “Boundary Analysis” for continuous algebraic expressions.

Core Concepts

• Boundary Points (Cut Points): The values of $x$ where the function can change sign. For polynomials, these are the real zeros of the function.
• Interval Testing: The real number line is divided into intervals by the boundary points. Because of the Intermediate Value Theorem, the function’s sign remains constant within each interval.
• Strict vs. Non-Strict:
  • $> or <$: Boundary points are excluded (open circles/parentheses).
  • $\ge or \le$: Boundary points are included (closed circles/brackets).
  • How to read: “The strict inequalities greater than or less than; and the non-strict inequalities greater than or equal to, or less than or equal to.”
  • Meaning / when to use: Include real zeros for $\ge/\le$; exclude them for strict inequalities.

Connected Concepts

• Linear Inequalities
• Interval Notation
• Polynomials: Basic Operations
• Rational Inequalities