Polynomial and rational inequalities: sign analysis, critical points and excluded values

What this dot point is asking

NESA wants you to solve polynomial inequalities like $P(x) \ge 0$ and rational inequalities like $\frac{P(x)}{Q(x)} < 0$ by analysing the sign of the expression across the intervals carved out by its critical values.

The answer

The sign-table method

1. Factor the polynomial or rational expression completely.
2. Identify critical values: where each factor equals zero (sign-change candidates) and, for rational expressions, where the denominator is zero (excluded values).
3. Order the critical values on a number line.
4. Determine the sign of each factor in each interval.
5. Multiply the signs to get the sign of the whole expression.
6. Combine intervals based on the direction of the inequality and the type of endpoint (open or closed).

The whole method rests on a single idea: a product (or quotient) of factors can only switch sign where one of the factors is zero or undefined. Between consecutive critical values nothing can flip, so the sign of the expression is constant on each interval, and you can read it off from a single test point. That is why the line is carved at the critical values and tested interval by interval.

Critical values

A polynomial $P(x)$ changes sign across each root of odd multiplicity, and does not change sign across roots of even multiplicity.

A rational expression $\frac{P(x)}{Q(x)}$ has critical values at the zeros of both $P$ and $Q$. Zeros of $Q$ are excluded from the solution because the expression is undefined there.

Handling multiplicity

A factor like $(x - 2)^2$ touches the x-axis but does not cross. The sign is the same either side. Treat it as not changing the sign of the product.

A factor like $(x - 2)^3$ behaves like $(x - 2)$ for sign purposes: sign changes across $x = 2$.

The shortcut: for sign analysis, count multiplicities modulo $2$. Even multiplicities can be ignored.

Rational inequalities: never multiply by an expression of unknown sign

A common rookie error is to multiply both sides of $\frac{A}{B} \ge 0$ by $B$ to clear the denominator. This is invalid because $B$ might be negative. Instead, move everything to one side and analyse the sign of the rational expression as a single object.

If you must clear the denominator, multiply by $B^2$ (always non-negative) instead. Or split into cases based on the sign of $B$. The sign-table method avoids these pitfalls.

Quadratic inequalities

For $a x^2 + b x + c \ge 0$ with $a > 0$, the parabola opens upward.

• If $\Delta > 0$ (two real roots $\alpha < \beta$): expression is positive outside the roots, that is $x \le \alpha$ or $x \ge \beta$.
• If $\Delta = 0$: expression is $\ge 0$ everywhere.
• If $\Delta < 0$: expression is positive everywhere.

For $a < 0$, all signs flip.

Endpoint conventions

• Strict inequality ($>$ or $<$): exclude critical values where the expression equals zero. Always exclude where the expression is undefined.
• Weak inequality ($\ge$ or $\le$): include zeros of the numerator, exclude zeros of the denominator (still undefined).

Build the sign diagram, stage by stage

A sign diagram is the cleanest way to show your working, and markers can follow it at a glance. Below, the inequality $x^3 - 4x \ge 0$ is solved by building the sign diagram one stage at a time. Factor first: $x^3 - 4x = x(x - 2)(x + 2)$, so the critical values are $x = -2, 0, 2$.

Stage 1, mark the critical values on a number line. Each factor gives one zero: $x + 2 = 0$ at $x = -2$, $x = 0$, and $x - 2 = 0$ at $x = 2$. Plot the three values; they cut the line into four intervals to test. (Draw them as open circles for now; whether they end up included depends on the inequality sign, decided at the end.)

Stage 2, find the sign of each factor in every interval. Pick a convenient test point inside each interval, here $x = -3, -1, 1, 3$, and record whether each factor is positive or negative. A linear factor $(x - r)$ is negative to the left of its own zero $r$ and positive to the right, so each row is a clean run of minuses then pluses.

Stage 3, multiply the columns to get the product row. In each interval, multiply the three factor signs together: three negatives give a negative, two negatives and a positive give a positive, and so on. Reading left to right the product runs $-, +, -, +$. Because every root here is simple (multiplicity $1$), the sign alternates at every critical value; a double root would break that alternation by repeating a sign.

Stage 4, read off the solution and fix the endpoints. The inequality is $\ge 0$, so keep the intervals where the product is positive, $(-2, 0)$ and $(2, \infty)$, and because it is a weak inequality, include the zeros $-2, 0, 2$ where the expression equals $0$. Shade those intervals with closed endpoints.

The solution is $-2 \le x \le 0$ or $x \ge 2$, that is $x \in [-2, 0] \cup [2, \infty)$. (This is the same answer as the cubic worked example below, reached via the diagram instead of a table; either presentation earns full marks.)

How exam questions ask about inequalities

The wording signals which type of sign analysis is wanted:

• "Solve $P(x) > 0$" or "find all $x$ such that..." Factor, build a sign diagram or table, and give the answer in interval notation. The direction of the inequality and the strict-versus-weak distinction control the endpoints.
• "Solve $\frac{P(x)}{Q(x)} \ge 0$" (a rational inequality). Do not cross-multiply. Move everything to one side, combine over a common denominator, then sign-analyse numerator and denominator together. Exclude every zero of the denominator, even under $\ge$.
• "Solve $\frac{A}{B} > c$" with a constant on the right. Subtract $c$ first to get $\frac{A}{B} - c > 0$, combine into a single fraction, then analyse. Multiplying both sides by $B$ is the classic trap because $B$'s sign is unknown.
• "For what values of $x$ is the curve above the line $y = g(x)$?" Rewrite as $P(x) - g(x) > 0$ and sign-analyse the difference. "Above" means the difference is positive; "below" means negative.
• "Hence solve [inequality]" after a factorise or graph part. Use the factorisation or sketch you just produced; the sign diagram is often just reading the graph's position relative to the x-axis.