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Math101learn.math101.caPolynomial Inequalities
Polynomial inequalities are solved by locating zeros and determining where the polynomial is positive or negative.
Zeros divide the number line into intervals where a continuous polynomial keeps a consistent sign.
Standard form first
Move every term to one side so the inequality compares a polynomial with zero:
This turns the question into: where is the graph above, on, or below the $x$-axis?
Critical numbers
Factor $P(x)$ and find its real zeros. These zeros are the only places where a polynomial can change sign, because polynomials are continuous.
Place the zeros in order on a number line. They divide the domain into test intervals. Within any interval containing no zero, the sign cannot change.
Sign-chart method
For each interval, either substitute one convenient test value into the original/factored expression or determine the sign of each factor. Record the product's sign and select intervals matching the inequality.
Testing one point per interval is sufficient because continuity prevents an unseen sign change without another zero.
Worked example: a quadratic inequality
Strict versus inclusive endpoints
Use closed endpoints for $\le$ or $\ge$ when a zero makes the expression equal to zero. Use open endpoints for $<$ or $>$ because equality is excluded.
Interval notation makes this distinction visible: brackets include finite endpoints; parentheses exclude them.
Multiplicity and sign changes
At a zero of odd multiplicity, the polynomial changes sign. At a zero of even multiplicity, it touches zero but keeps the same sign.
For
the sign changes at $x=-1$ but not at $x=2$. Multiplicity can speed up a sign chart, though one test point remains a reliable check.
Worked example: higher degree
Solve
Zeros are $-2$, $1$ (even multiplicity), and $4$. Testing or tracking factor signs gives positive intervals $(-\infty,-2)$ and $(4,\infty)$; the sign does not change at $1$.
Because the inequality is strict, none of the zeros are included:
Graphical solution
Graph $y=P(x)$ and identify where it lies above or below the $x$-axis. This provides an excellent check and can estimate irrational zeros.
For an exact algebraic question, a graphing window alone may miss tangencies or distant roots, so use structure and factorization whenever possible.
Inequalities from two functions
To solve $f(x)\ge g(x)$, rearrange to
Zeros of the difference are intersection inputs. The sign chart then identifies where the graph of $f$ lies above or on the graph of $g$.
Context restrictions
A polynomial model may apply only for time $t\ge0$, a finite production interval, or whole-number quantities. Intersect the algebraic solution with the contextual domain.
Report the meaning of boundary values: they may represent break-even points, threshold times, or equality cases.
Common mistakes
Solving only the related equation. Zeros are boundaries, not the complete inequality solution.
Assuming the sign alternates at every zero. Even multiplicity does not change sign.
Including endpoints in a strict inequality. Check the symbol.
Testing exactly at a zero. Use interior points to determine interval signs.
Ignoring the original domain. Context may remove part of the algebraic solution.
Quick self-check
- Is one side equal to zero?
- Are all real zeros found and ordered?
- Have multiplicities been recorded?
- Is each interval's sign justified?
- Do brackets or parentheses match the inequality?
- Has the result been intersected with the contextual domain?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve x² − 2x − 3 ≤ 0.
- The zeros are −1 and 3.
- The upward-opening quadratic is nonpositive between its zeros.
- Include both endpoints: [−1, 3].
End of lesson
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