The Zero-Product Property

≈ 25 min

The bridge between factorising and solving is one deceptively simple fact:

\text{If } A \times B = 0, \text{ then } A = 0 \text{ or } B = 0.

No other number has this property — if $A \times B = 12$ , you know almost nothing about $A$ or $B$ individually ( $2 \times 6$ ? $3 \times 4$ ? $\frac{1}{2} \times 24$ ?). But a zero product forces a zero factor.

Worked example. Solve $(x - 2)(x + 5) = 0$ .

Either $x - 2 = 0$ , giving $x = 2$ , or $x + 5 = 0$ , giving $x = -5$ .

Two factors, two candidate roots — and both check: $(2-2)(2+5) = 0 \times 7 = 0$ ✓, $(-5-2)(-5+5) = -7 \times 0 = 0$ ✓

If $A \times B = 0$ , what can you conclude?

$A = 0$ and $B = 0$ (both must be zero)

$A = 0$ or $B = 0$ (at least one is zero)

$A = B$

Nothing — many pairs multiply to zero

Solve: $(x - 2)(x + 5) = 0$ . Answer in the form x=… or x=…

What is the smaller root of $(x - 1)(x - 6) = 0$ ?