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Factoring

Factoring is multiplication read backwards, so every identity below can be checked by expanding it again — and you should, every time. The skill is not memorising nine patterns; it is looking at an expression and knowing which one it is before you start writing. That recognition step is what the worked solutions lead with, and each example here names the tell before it names the move.

Factoring exercises with solutions

Calculator
The order of attack
Work down this list every time. Most mistakes are not arithmetic — they are starting at step three when step one was still available.

  1. 1. Greatest common factor, always first. Take out everything every term shares: the numeric GCF and the lowest power of each common letter. A GCF pulled out first often turns what is left into a pattern you can already see. Skipping it is the single most common reason an answer is marked incomplete.

  2. 2. Count the terms. Two terms: look for a difference of squares, or a sum or difference of cubes. Three terms: a perfect square trinomial or an ordinary trinomial. Four terms: grouping. The count tells you which shelf to reach for.

  3. 3. Look for a substitution. If the exponents come in a -and- pattern ( and , and ) or the same binomial appears twice, set equal to the repeating part. The problem collapses to one you have already solved.

  4. 4. Factor completely. Look at every factor you produced and ask whether it factors again. is not finished at , and is not finished either.

  5. 5. Check by expanding. Multiply your factors back out. This is not optional politeness — it catches every sign error, and it is exactly how the worked solutions verify themselves.

The identities
Learn each one in both directions. Left to right factors; right to left expands, which is how you check.
  • Common factor

    The distributive law, used backwards. Works for any number of terms.

  • Difference of squares

    Two terms, both perfect squares, separated by a minus. The middle terms cancel when you expand, which is why nothing is left over.

  • Sum of squares

    There is no identity for it. Recognising it as unfactorable is itself the answer — unless a trick term can be added and subtracted (see below).

  • Perfect square trinomial

    Three terms, first and last are perfect squares, middle is exactly twice the product of their roots.

  • Perfect square trinomial, negative

    Same shape with a minus in the middle. The sign of the middle term is the sign inside the bracket.

  • Sum of cubes

    Signs read Same, Opposite, Always Positive. The trinomial never factors further.

  • Difference of cubes

    Same pattern, same sign rule. Only the binomial's sign changes.

  • Grouping

    Four terms: split into two pairs, take a GCF from each, and the leftover brackets must match exactly. If they do not, try re-pairing.

  • Simple trinomial

    Leading coefficient 1: find two numbers that multiply to the constant and add to the middle coefficient.

  • General trinomial

    The ac method. Splitting the middle term turns any trinomial into a grouping problem.

Common factors
Nothing else is attempted until this is done.

Two terms — squares and cubes
With two terms there are only three possibilities: difference of squares, sum of cubes, difference of cubes. A sum of squares is none of them.

Three terms — trinomials
Check for a perfect square first: it is the fastest case and it is easy to miss once you have started hunting for factor pairs.

Four terms — grouping
Split into two pairs, take a GCF from each, and the brackets left behind must be identical. If they are not, either the sign is wrong or the pairs are.

Substitution
When the exponents come in a -and- pattern, or a binomial repeats, name the repeating part and the problem becomes an ordinary trinomial. Substitute back at the very end — and then check whether the result factors further.

Where this goes wrong
Each of these is a real slip from real working, not a hypothetical.

  • A sum of squares does not factor over the reals at all. Expanding shows the missing : test at , where .

  • The has to come along. expands to — no anywhere. Always expand the square you wrote and compare it term by term.

  • The middle sign of the trinomial is always opposite the binomial's. Same, Opposite, Always Positive — and the reason is the cancellation: only with do the and terms disappear on expanding.

  • Correct as far as it goes, but not factored completely. Re-inspect every factor you produce before calling it done.

  • Both are true, but only the second has matching brackets, which is the point of grouping. When the second bracket comes out backwards, factor out the negative instead.

  • Dividing a term by the GCF leaves 1, not nothing. Expanding gives and loses the entirely.

Practice
Factor completely. GCF first, then count the terms.

Type an answer and it gets marked. What you get right is kept on this device.