Polynomial Inequalities Using Sign Charts

Strategy: find roots, build sign chart, test intervals

For any polynomial inequality in factored form, the roots divide the number line into intervals. The polynomial's sign can only change at a root, so test one point per interval.

Step-by-step: Solve $(x - 1)(x + 2)(x - 4) > 0$

Step 1 — Find the roots: $x = 1$, $x = -2$, $x = 4$.

Step 2 — Order the roots and mark intervals:

$(-\infty, -2), \quad (-2, 1), \quad (1, 4), \quad (4, \infty)$

Step 3 — Build the sign chart. Test one value from each interval:

• $x = -3$: $(-3-1)(-3+2)(-3-4) = (-4)(-1)(-7) = -28 < 0$ → negative
• $x = 0$: $(0-1)(0+2)(0-4) = (-1)(2)(-4) = +8 > 0$ → positive
• $x = 2$: $(2-1)(2+2)(2-4) = (1)(4)(-2) = -8 < 0$ → negative
• $x = 5$: $(5-1)(5+2)(5-4) = (4)(7)(1) = +28 > 0$ → positive

Step 4 — Select intervals where the product is positive (since we want $> 0$):

$x \in (-2, 1) \cup (4, \infty)$

Step 5 — Check boundaries: Since the inequality is strict ($>$, not $\geq$), the roots themselves are excluded (open circles).

The alternating sign pattern

For a polynomial with all simple roots (each root appears once) and positive leading coefficient, the signs alternate: $-, +, -, +, \ldots$ starting from the rightmost interval as $+$. This shortcut saves time on exams.

Try it in the visualization

Adjust the three roots. The sign chart shows $+$ and $-$ in each interval. The cubic graph confirms which regions are above/below the x-axis. The solution set is highlighted.