Factoring completely means breaking down an algebraic expression into its simplest prime factors so that no sub-expression can be factored further. If you stop after a single step, you haven't finished the job.
To crack any factoring problem, you must follow a strict sequential hierarchy: always check for a Greatest Common Factor (GCF) first, look for special binomial or trinomial patterns next, and apply advanced grouping or coefficient methods last. Missing just one hidden factor leaves the expression incomplete and points on an exam lost.
The Non-Negotiable First Rule: Always Extract the GCF
Before you try any quadratic tricks or grouping methods, look at every term in the expression. Is there a common numerical factor, a shared variable, or both that divides evenly into every term?
Factor out that GCF immediately. If you skip this, your subsequent binomials will carry hidden multipliers that make further reduction impossible.
Consider the expression $3x^3 - 15x^2 + 18x$.
- Look at the coefficients: $3$, $15$, and $18$ share a greatest common factor of $3$.
- Look at the variables: $x^3$, $x^2$, and $x$ share an $x$.
- Pull out the complete GCF, which is $3x$.
- Divide each term by $3x$ to find the remaining polynomial: $3x(x^2 - 5x + 6)$.
You aren't done yet, but pulling $3x$ out simplifies the interior quadratic expression significantly.
Spotting Special Products
Once you isolate the core polynomial, scan for recognizable patterns. Recognizing these structures saves you from tedious trial-and-error factoring.
Difference of Squares
If you have a binomial where two squared terms are separated by a minus sign, use the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$.
For example, $9x^2 - 25$.
- The first term is $(3x)^2$, so $a = 3x$.
- The second term is $(5)^2$, so $b = 5$.
- Factored completely, it becomes $(3x - 5)(3x + 5)$.
Never try this rule on a sum of squares like $x^2 + 25$. Over real numbers, sums of squares cannot be factored.
Perfect Square Trinomials
When a three-term quadratic has an outer first and last term that are squares, check if it fits a perfect square pattern: $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
For example, $4x^2 - 12x + 9$.
- The first term is $(2x)^2$.
- The last term is $(3)^2$.
- The middle term equals $2(2x)(3) = 12x$.
- Because the middle term matches, it factors into $(2x - 3)^2$.
Factoring Standard Trinomials
When a trinomial isn't a perfect square, you must break it down into two binomials. The strategy depends on the leading coefficient.
When the Leading Coefficient is One ($x^2 + bx + c$)
You need two integers that multiply to give $c$ and add up to give $b$.
Take $x^2 - 7x + 12$.
- You need numbers that multiply to $12$ and add up to $-7$.
- Those numbers are $-3$ and $-4$.
- Write the factored form: $(x - 3)(x - 4)$.
When the Leading Coefficient is Greater Than One ($ax^2 + bx + c$)
When $a$ is not $1$, use the grouping method (often called the AC method).
Take $2x^2 + 7x + 3$.
- Multiply $a$ and $c$: $2 \times 3 = 6$.
- Find two factors of $6$ that add up to the middle coefficient, $7$. Those factors are $6$ and $1$.
- Split the middle term using those factors: $2x^2 + 6x + 1x + 3$.
- Factor by grouping in pairs: $2x(x + 3) + 1(x + 3)$.
- Factor out the common binomial: $(2x + 1)(x + 3)$.
Tackling Four Terms with Factoring by Grouping
If an expression contains four or more terms and doesn't fit standard quadratic forms, group the terms into pairs to find a common binomial factor.
Consider $x^3 + 4x^2 - 9x - 36$.
- Group the first two terms and the last two terms: $(x^3 + 4x^2) - (9x + 36)$.
- Factor the GCF out of each group: $x^2(x + 4) - 9(x + 4)$.
- Notice the shared binomial $(x + 4)$. Factor it out: $(x^2 - 9)(x + 4)$.
- Check your work: the term $(x^2 - 9)$ is a difference of squares and factors further into $(x - 3)(x + 3)$.
- The completely factored expression is $(x - 3)(x + 3)(x + 4)$.
Stopping at $(x^2 - 9)(x + 4)$ is a classic mistake that leaves the polynomial only half-factored.
Common Mistakes to Avoid
Even experienced math students trip up on subtle factorization traps. Keep these pitfalls in mind:
- Forgetting the GCF: Leaving a common factor inside the parentheses means your expression isn't completely reduced. Always pull constants and variables out first.
- Stopping too early: Assuming an expression is done because it looks complex. Always check remaining binomials or trinomials for further squares or factorable pairs.
- Distributing minus signs incorrectly: When factoring out a negative GCF, remember to flip the signs of every interior term.
- Treating sums of squares as factorable: Remember that $x^2 + y^2$ cannot be broken down using real numbers.
Frequently Asked Questions
How do I know when an expression is factored completely?
An expression is factored completely when every remaining factor is prime. No individual binomial or trinomial can be broken down into simpler polynomial factors with integer coefficients.
What should I do if a polynomial doesn't factor?
If you've tested all standard methods—GCF, grouping, special products, and trial trinomial splitting—and nothing works, the polynomial is considered irreducible over the integers. You can check out algebraic resources like Khan Academy Algebra for practice problems on prime polynomials.
Can I check my answer by multiplying it back out?
Yes. Expansion is the ultimate safety net. Multiply your final factored binomials together using the distributive property or FOIL method. If the result matches your original starting expression precisely, your factorization is correct.
Does the order of the factors matter?
Multiplication is commutative, so writing $(x - 2)(x + 3)$ is identical to writing $(x + 3)(x - 2)$. However, standard convention places monomial GCFs at the very front, followed by binomials in increasing or alphabetical order of variables.
