Factoring: rewriting an expression into an equivalent form

$3x + 6$ and $3(x + 2)$ are the same expression. The first is a sum; the second is a product. Rewriting the first as the second is factoring. It is the rewrite skill in the substitutions exercises: rewriting an expression into an equivalent form that exposes structure hidden by the original. Factoring is substitution’s quiet sibling. Substitution fills a value in; factoring pulls a factor out.

Factoring is the distributive law, run backwards

The skill uses one law: the distributive law, read in reverse. Read forwards, it says a multiplier distributes over a sum:

$$a(b + c) = ab + ac$$

Read backwards, it says a sum with a common factor can be rewritten as a product:

$$ab + ac = a(b + c)$$

That backwards reading is factoring. Look at $3x + 6$. Both terms share a factor of $3$, because $6$ is $3 \cdot 2$, so pull it out front: $3(x + 2)$. The expression has not changed - the two forms are equivalent for every $x$ - but the product form shows immediately that $x = -2$ makes it zero. The sum form hides that fact.

The three shapes to recognize

The rewrite exercises use three recognizable shapes, in increasing difficulty.

A common factor. $3x + 6$ and $2a + 2b$ both factor by pulling out the shared factor: $3(x + 2)$ and $2(a + b)$. Decide what the common factor is, and whether it should be negative. $-3x - 6$ factors to $-3(x + 2)$, because factoring out the negative leaves both remaining terms positive and clean.

A perfect square. $x^2 + 2x + 1$ is the square of a binomial: it factors to $(x + 1)(x + 1)$. The pattern is “first term squared, twice the product of the two, last term squared”: here $x^2$, then $2 \cdot x \cdot 1$, then $1 \cdot 1$. Reverse that pattern into $(x + 1)$ written twice.

A difference of squares. $x^2 - 9$ factors to $(x + 3)(x - 3)$, because any difference of squares $a^2 - b^2$ is $(a + b)(a - b)$. The outer terms multiply to $a^2 - b^2$, and the middle terms cancel. It is the same identity the equivalence article cites in the other direction - here you produce the factored form rather than recognize it.

Why the factored form matters

A factored form is not “more correct” than the expanded form. They are equivalent. It is more informative. $3(x + 2)$ tells you the value that makes the expression zero; $3x + 6$ buries that fact in a sum. A perfect square tells you the expression is never negative, because a square is never negative. The expanded form makes you work that out. The skill is not “get the right answer.” It is choosing the form that makes the next question easy - the same instinct as refactoring code so its intent is visible in its shape.

That connects factoring to the wider site. “Rewrite into an equivalent form” is the mathematical version of “change the shape, keep the meaning,” which is what refactoring is. Equivalence makes the rewrite safe: the two forms agree for every value, so you can rewrite without re-checking.

Why this matters here

The rewrite exercises in the substitutions set ask for exactly this reverse-distribution, one shape at a time. The oracle grades the form (an equivalent product), not any particular spelling. That is the point: there is usually one clean factored form, and producing it is the skill. Once you can factor fluently, the equivalence question gets easier too. Two expressions are equivalent when they factor to the same thing, or when one is the other with a law applied.

Where to go next

• Equivalence: are two expressions the same for every value?
  • the guarantee that makes rewriting safe, and the question factoring answers.
• Substitution: replace a name with a value
  • the forward move factoring runs backwards from.
• The laws hidden in your code
  • the distributive law, and its siblings, as the rules that make code safe to rearrange.