Factoring

Also called: factorization, factorisation, factor out, factored form, factorized form

Pulling a literal or term shared by several products out in front of a bracket, as in AB + AC = A(B + C). It is the distributive law read backwards.

Factoring rewrites as : a literal that appears in several terms is written once, outside a bracket. It is the distributive law used right to left.

It is useful for three reasons:

It reveals simplifications. factors to , and the complement law turns the bracket into 1. That is how combining terms works.
It can save hardware. has 4 literals; has 3. A factored form is a multi-level circuit, which can use fewer gate inputs than the two-level SOP, though signals may pass through more gates.
It is a step in many proofs, such as absorption: = = A.

You can factor out a whole product too: = .

There is also an OR version: = factors A out of two sums; see or over and.

Worked examples

Example

Factoring a shared literal

Factor .

1.
Both terms contain .
2.
Write once and put what is left in a bracket: .
3.
Check by multiplying out: . ✓
4.
Literal count drops from 4 to 3.

Example

Factoring to simplify

Simplify .

1.
Factor out : .
2.
Inside the bracket, = (redundant literal rule).
3.
Result: , or multiplied out, .

Common mistakes

Factoring a literal that is not in every term of the group, such as writing as .
Forgetting that a factored form is not an SOP. If the question asks for a sum of products, multiply back out.

Practice Factoring

Interactive questions with instant feedback and a worked solution for every wrong answer.

Boolean Algebra lesson

Factor .

Learn it step by step

Factoring is taught in Boolean Algebra.

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