Polynomial Inequalities

Polynomial inequalities are solved by locating zeros and determining where the polynomial is positive or negative.

Interpret → represent → solve → check. The problem is translated into a mathematical representation, solved with an appropriate operation, and checked against the original quantities and conditions.

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:

P(x)>0,quad P(x)\ge0,quad P(x)<0,\quad\text{or}\quad P(x)\le0.

This turns the question into: where is the graph above, on, or below the $x$-axis?

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

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

$$ P(x)=(x-2)^2(x+1), $$

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

$$ (x+2)(x-1)^2(x-4)>0. $$

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:

(-\infty,-2)\cup(4,\infty).

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.