Theorem

A product is zero only when a factor is zero

Algebra • Equations

Formula

A B = 0 ⟺ A = 0 or B = 0

For real numbers,

A B = 0

if and only if

A = 0

B = 0

; an equation should therefore be brought to the form

A B = 0

and factorised, and never divided through by an expression that may itself be zero.

How to use it

1Move everything to one side, factorise, then set each factor to zero in turn.
2When the same bracket appears on both sides, collect it rather than cancelling it: cancelling discards the root it supplies.
3Check each root against any restriction the original equation carried, such as a denominator that must not vanish.

Solve

x (x - 5) = 3 (x - 5)

Step-by-step

1
$x (x - 5) - 3 (x - 5) = 0$
Why: Collect on one side instead of dividing by $x - 5$ .
2
$(x - 5) (x - 3) = 0$
Why: The common bracket factors out.
3
$x = 5$ or $x = 3$
Why: Cancelling $x - 5$ at the start would have lost the root $x = 5$ .

Answer

x = 3

x = 5

Watch out

The property is about zero only:

A B = 6

says nothing at all about

A

B

separately. Dividing both sides by a bracket silently assumes it is non-zero and throws away a root.

Also known as

zero product rule, null factor law, set each factor to zero, do not cancel a common factor