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Polynomial Inequality

A polynomial inequality is an inequality that contains a polynomial expression on one or both sides, such as x2−4x+3>0x^2 - 4x + 3 > 0. You solve it by finding the zeros of the polynomial and then testing intervals to determine where the inequality is true.

A polynomial inequality is a mathematical statement that compares a polynomial expression to a value (often zero) using an inequality symbol (<<, >>, ≤\leq, or ≥\geq). To solve one, you first rearrange the inequality so that one side is zero, then factor the polynomial to find its real zeros. These zeros divide the number line into intervals, and you test each interval to determine where the polynomial satisfies the inequality. The solution is the union of all intervals where the inequality holds, including or excluding the zeros depending on whether the inequality is strict or non-strict.

Worked Example

Problem: Solve the inequality x2−x−6≤0x^2 - x - 6 \leq 0.
Step 1: Factor the polynomial: Factor the left side of the inequality.
x2−x−6=(x−3)(x+2)≤0x^2 - x - 6 = (x - 3)(x + 2) \leq 0
Step 2: Find the zeros: Set each factor equal to zero. The zeros are x=3x = 3 and x=−2x = -2. These divide the number line into three intervals: (−∞,−2)(-\infty, -2), (−2,3)(-2, 3), and (3,∞)(3, \infty).
x−3=0⇒x=3x+2=0⇒x=−2x - 3 = 0 \Rightarrow x = 3 \qquad x + 2 = 0 \Rightarrow x = -2
Step 3: Test each interval: Pick a test point in each interval and evaluate the sign of (x−3)(x+2)(x-3)(x+2).
x=−3:  (−6)(−1)=+6>0x=0:  (−3)(2)=−6<0x=4:  (1)(6)=+6>0x = -3:\;(-6)(-1) = +6 > 0 \qquad x = 0:\;(-3)(2) = -6 < 0 \qquad x = 4:\;(1)(6) = +6 > 0
Step 4: Identify where the inequality holds: The product is negative (less than zero) only on the interval (−2,3)(-2, 3). Since the inequality uses ≤\leq, also include the endpoints where the expression equals zero.
Answer: The solution is −2≤x≤3-2 \leq x \leq 3, or in interval notation, [−2, 3][-2,\, 3].

Visualization

Why It Matters

Polynomial inequalities appear whenever you need to find where a quantity stays above or below a certain value. In physics, you might determine the time intervals during which a projectile is above a given height. In economics, they help identify ranges where profit exceeds a threshold. Mastering sign analysis also builds the foundation for working with rational inequalities and analyzing function behavior in calculus.

Common Mistakes

Mistake: Solving as if it were an equation and writing only the zeros as the answer.
Correction: The zeros are just boundary points. You need to test the intervals between them to find all xx-values that satisfy the inequality, then express the solution as an interval or union of intervals.
Mistake: Including or excluding endpoints incorrectly.
Correction: For strict inequalities (<< or >>), use open endpoints — the zeros themselves are not part of the solution. For non-strict inequalities (≤\leq or ≥\geq), include the zeros with closed brackets.

Related Terms

  • Inequality — General concept that polynomial inequalities extend
  • Polynomial — The type of expression used in these inequalities
  • Zero of a Function — Boundary points found when solving the inequality