Math101 · Independent Ontario learning library
Math101learn.math101.caFactoring
Factoring rewrites an expression as a product, revealing common structure, zeros, simplifications, and efficient algebraic methods.
Factoring rewrites a sum or difference as an equivalent product. It is reverse distribution.
Why product form matters
Product form exposes repeated factors and zeros. $x^2-5x+6$ does not visibly show its roots, but
shows that the expression is zero at $x=2$ or $x=3$.
Always check the GCF first
Find the greatest numerical and variable factor shared by every term:
Factoring is incomplete if a nontrivial common factor remains inside brackets. Include a negative GCF when it makes the leading term inside positive.
Difference of squares
Two perfect squares separated by subtraction follow
Thus $9x^2-25=(3x-5)(3x+5)$. A sum of squares does not factor this way over the real numbers.
Trinomials
For $x^2+bx+c$, find numbers whose product is $c$ and sum is $b$. For $x^2+7x+12$, the numbers $3$ and $4$ give
When the leading coefficient is not $1$, use product-sum reasoning with $ac$, decomposition, or systematic trial.
Factoring by grouping
Group terms so each pair reveals a common binomial:
If the grouped binomials do not match, rearrangement or a sign adjustment may help.
A complete example
Continue until every factor is irreducible over the number system in use.
Solving with zero-product property
If a product equals zero, at least one factor is zero:
This works only after the equation is written with zero on one side. From $2x(x-2)(x+2)=0$, solutions are $x=0,2,-2$.
Verifying
Expand the factors and compare with the original expression. This check catches sign errors and incomplete middle terms. A few substitutions can help, but expansion verifies equivalence for all $x$.
Choosing a strategy
Use this scan:
- GCF?
- Two terms: difference of squares or cubes?
- Three terms: trinomial or perfect square?
- Four terms: grouping?
- Can substitution reveal a pattern such as quadratic in $x^2$?
Common mistakes
Skipping the GCF. Later patterns may look harder or remain incomplete.
Factoring a sum of squares as a difference. Signs will fail when expanded.
Using zero-product property before one side is zero. A product equal to another number does not split directly.
Stopping too early. Inspect every new factor again.
Quick self-check
- Is there a GCF?
- Which term-count or special-product pattern appears?
- Is each factor fully reduced?
- Does expanding reproduce every original term?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Factor 2x³ − 8x completely.
- 2x³ − 8x = 2x(x² − 4)
- x² − 4 = (x − 2)(x + 2)
- The complete factorization is 2x(x − 2)(x + 2).
End of lesson
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