Solve polynomial and rational inequalities by factoring and analysing the sign of each factor across critical 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).

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).