Sign table: the method that organizes everything

Why this table

To solve $f(x) > 0$ (or $< 0$, $\geq 0$, etc.), we ask on which intervals $f(x)$ takes which sign. The table lays that out cleanly.

The method in 4 moves

1. Factor $f(x)$ (if not already) into a product or quotient of simple factors.
2. Find the roots of each factor (values of $x$ where the factor vanishes).
3. Draw the table: one row per factor, one column between each root, marking zeros at roots.
4. Multiply signs row by row to get the sign of $f$.

Example: sign of $f(x) = (x - 2)(x + 3)$

Roots: $x = 2$ and $x = -3$.

$x$	$-\infty$		$-3$		$2$		$+\infty$
$x - 2$		$-$	$-$	$-$	$0$	$+$
$x + 3$		$-$	$0$	$+$	$+$	$+$
$f(x)$		$+$	$0$	$-$	$0$	$+$

So $f(x) > 0$ on $]-\infty; -3[ \cup ]2; +\infty[$.

Sign rules reminder

$(-) \times (-) = (+)$. $(-) \times (+) = (-)$. Two negatives → positive.

For a quotient

The sign of $\dfrac{u}{v}$ follows the same rule as $u \times v$. Careful: values that cancel the denominator are forbidden (double bar in the table).

?Your turn

The sign of f(x) = (x − 1)(x + 4) on ]−4; 1[ is:

A.PositiveB.NegativeC.ZeroD.Undefined

Check