Factoring Polynomials: The Reverse of Multiplying, Made Systematic
Factoring is just multiplication run backward — turning a sum into a product. A short checklist of patterns, tried in the right order, factors almost anything the SAT, ACT, or SSAT can throw at you.
The Short Version
- Factoring rewrites a sum as a product — the reverse of using the distributive property.
- Always pull out the greatest common factor (GCF) first; it simplifies everything that follows.
- Know the patterns: difference of squares (a² − b²), trinomials (x² + bx + c), and grouping for four terms.
- Factoring is the fastest way to solve quadratics — central to the SAT, ACT, and SSAT.
When you multiply (x + 2)(x + 3) you get x² + 5x + 6. Factoring just runs that process backward: start with x² + 5x + 6 and recover (x + 2)(x + 3). Why bother? Because a product equal to zero is incredibly easy to solve — if two things multiply to zero, one of them must be zero. That single fact is why factoring unlocks quadratics, rational expressions, and a surprising number of test questions.
This guide gives you the patterns in the order you should try them, shows where each test uses factoring, and ends with worked and practice problems matched to real test difficulty at Northside Tutoring.
Why Factoring Matters
Factoring is a gateway skill. It's the fastest way to solve quadratics, it's required to simplify rational expressions, and it powers questions about a polynomial's zeros and graph. On the SAT especially, recognizing a factorable form — rather than grinding through the quadratic formula — can save a minute per question.
Step 1: Always Pull the GCF
Before anything else, factor out the greatest common factor — the largest number and variable shared by every term. It makes the remaining expression smaller and often reveals a familiar pattern:
Skipping this step is the most common reason students get stuck.
The Difference of Squares
Any expression of the form a² − b² (a perfect square minus a perfect square) factors instantly:
So x² − 25 = (x + 5)(x − 5). Note there is no such factoring for a sum of squares over the real numbers — a² + b² does not factor. The tests check whether you know the difference.
Factoring Trinomials
For x² + bx + c, find two numbers that multiply to c and add to b. Those two numbers become the constants in your binomials:
because 3 × 4 = 12 and 3 + 4 = 7. When the leading coefficient isn't 1 (like 2x² + 7x + 3), use grouping or trial-and-error with the factor pairs.
The sign road map
If c is positive, both numbers share b's sign. If c is negative, the two numbers have opposite signs, and the larger one carries b's sign. This narrows the search instantly.
Factoring by Grouping
When you have four terms, group them in pairs, factor each pair, and look for a common binomial:
Grouping is also the reliable way to factor harder trinomials with a leading coefficient.
The Factoring Checklist
Try these in order, every time:
- GCF — pull out the greatest common factor.
- Two terms? Check for a difference of squares.
- Three terms? Factor the trinomial (multiply-to-c, add-to-b).
- Four terms? Factor by grouping.
- Always check whether a factor can be factored further.
Where You'll See This — Test by Test
Factoring is pure technique — nothing is provided on a reference sheet. It underlies the quadratics and rational-expression questions that appear all over the SAT and ACT, and the SSAT Upper Level introduces the basic patterns.
Digital SAT
Constantly used to solve quadratics and simplify expressions. Recognizing a factorable form is a major time-saver on the Digital SAT.
Explore SAT Tutoring → College AdmissionsACT
Tested directly and as a tool inside other problems. Difference of squares and trinomials are the most common.
Explore ACT Tutoring → Independent School AdmissionsSSAT
Upper Level introduces pulling a GCF and factoring simple trinomials.
Explore SSAT Tutoring → K-12 CurriculumSchool Algebra
A core Algebra I skill that recurs throughout Algebra II and pre-calculus.
Explore Algebra Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Factoring — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
For the developer / editor
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Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
Factor completely: 4x² − 16.
Show solution
Pull the GCF 4: 4(x² − 4).
x² − 4 is a difference of squares: 4(x + 2)(x − 2).
Factor: x² + 9x + 20.
Show solution
Two numbers multiplying to 20 and adding to 9: 4 and 5.
(x + 4)(x + 5).
Factor: x² − 36.
Show solution
Difference of squares with a = x, b = 6.
(x + 6)(x − 6).
Factor: x² − 5x − 14.
Show solution
Two numbers multiplying to −14 and adding to −5: −7 and +2.
(x − 7)(x + 2).
Factor by grouping: 2x³ + 6x² + 5x + 15.
Show solution
Group: 2x²(x + 3) + 5(x + 3).
Common binomial (x + 3): (x + 3)(2x² + 5).
Common Mistakes to Avoid
Three traps that catch students every year
- Skipping the GCF. Always pull common factors first; otherwise you'll wrestle with bigger numbers and may miss the final fully-factored form.
- Trying to factor a sum of squares. a² + b² does not factor over the real numbers. Only the difference of squares does.
- Stopping too early. After factoring once, check each factor again — 4(x² − 4) isn't done until it's 4(x + 2)(x − 2).
Practice Problems — You Try
Three problems below. Work each before checking the solution.
Factor: 3x² + 12x.
Show solution
GCF is 3x: 3x(x + 4).
Factor: x² − 11x + 28.
Show solution
Numbers multiplying to 28, adding to −11: −4 and −7.
(x − 4)(x − 7).
Factor completely: 2x³ − 18x.
Show solution
GCF 2x: 2x(x² − 9).
x² − 9 is a difference of squares: 2x(x + 3)(x − 3).
The Northside Method — How We Teach This 1-on-1
Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.
Every Northside student works through a four-step framework:
- Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
- Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
- Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
- Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.
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