Factoring in Algebra
Factoring is the process of rewriting a polynomial as a product of simpler expressions. It is the reverse of multiplication, and it is one of the most heavily tested skills in algebra — appearing in College Algebra, Algebra I, Algebra II, and as a prerequisite skill in precalculus and calculus. Mastering factoring means knowing which method to use for which type of polynomial, and knowing when your answer is fully factored.
The Five Factoring Methods
1. GCF — Always check this first, every time, no exceptions.
2. Difference of squares — a² − b² = (a+b)(a−b). Two terms, minus sign, both perfect squares.
3. Simple trinomial — x² + bx + c. Find two numbers that multiply to c and add to b.
4. AC method — ax² + bx + c where a ≠ 1. Multiply a × c, split the middle term, factor by grouping.
5. Grouping — Four-term polynomials. Group into pairs, factor GCF from each, factor out the common binomial.
Table of Contents
1) What Factoring Is and Why It Matters
Factoring reverses multiplication. When you multiply (x + 3)(x + 5), you get x² + 8x + 15. Factoring takes x² + 8x + 15 back to (x + 3)(x + 5). The factored form is useful because it reveals the structure of the polynomial — where it equals zero, how it behaves, and how it relates to other expressions.
Factoring is a prerequisite for solving quadratic equations, simplifying rational expressions, and working with polynomial functions. Most algebra courses spend more time on factoring than on any other single skill. On MyMathLab, WebAssign, ALEKS, and Hawkes Learning, factoring problems appear in nearly every homework module from mid-semester through the end of the course.
Choosing the right method
The biggest source of errors in factoring is not the algebra itself — it is applying the wrong method to a given polynomial. The decision flowchart below shows how to identify the correct method based on the number of terms and the form of the polynomial. The rule at the top of the chart is non-negotiable: always check for a GCF before doing anything else.
2) GCF — Always Check First
The greatest common factor (GCF) is the largest expression that divides evenly into every term of the polynomial. Factoring out the GCF first simplifies every subsequent step and is required for a complete factored form. On MyMathLab and ALEKS, an answer that has not had the GCF factored out is marked wrong even if the remaining factoring is correct.
How to find the GCF
Look at all the coefficients and find the largest integer that divides all of them. Then look at the variable part of each term and take the lowest exponent of each variable that appears in every term. Multiply these together to get the GCF.
Coefficients: GCF of 6, 9, 3 = 3
Variables: x appears in all terms, lowest power is x¹
GCF = 3x
6x³ + 9x² + 3x = 3x(2x² + 3x + 1)
After factoring out the GCF, check whether the remaining polynomial in parentheses can be factored further. In the example above, 2x² + 3x + 1 factors to (2x + 1)(x + 1), so the fully factored form is 3x(2x + 1)(x + 1).
GCF with negative leading coefficients
When the leading term is negative, factor out a negative GCF so the leading term inside the parentheses is positive. This is a MyMathLab and ALEKS convention that affects answer matching.
GCF = -4 (factor out negative to make leading term positive inside)
= -4(x² – 2x + 3)
For more on how MyMathLab handles algebraic answer entry: MyMathLab Help
3) Factoring Trinomials
Trinomials are three-term polynomials of the form ax² + bx + c. The method depends on whether the leading coefficient a equals 1 or not.
Case 1: a = 1 (simple trinomial)
For x² + bx + c, find two numbers p and q such that p × q = c and p + q = b. The factored form is (x + p)(x + q).
Need two numbers that multiply to 12 and add to 7.
Pairs that multiply to 12: (1,12), (2,6), (3,4)
Which pair adds to 7? → 3 and 4
x² + 7x + 12 = (x + 3)(x + 4)
Need two numbers that multiply to 6 and add to -5.
Both numbers must be negative (product positive, sum negative).
-2 × -3 = 6 and -2 + (-3) = -5 ✓
x² – 5x + 6 = (x – 2)(x – 3)
When c is negative, the two numbers always have opposite signs — one positive, one negative. The larger number takes the sign of b.
c is negative (-6), so one number is positive and one is negative.
Need two numbers that multiply to -6 and add to +1.
Pairs: (1,-6), (-1,6), (2,-3), (-2,3)
Which pair adds to 1? → -2 and 3 (-2 + 3 = 1 ✓)
x² + x – 6 = (x – 2)(x + 3)
Case 2: a ≠ 1 (AC method)
For ax² + bx + c where a ≠ 1, multiply a × c to get a target product. Find two numbers that multiply to that product and add to b. Rewrite the middle term bx using those two numbers, then factor by grouping. The step-by-step diagram below shows the full process.
a × c = 2 × 3 = 6
Find two numbers that multiply to 6 and add to 7: 1 and 6
Rewrite middle term: 2x² + 1x + 6x + 3
Group: (2x² + x) + (6x + 3)
Factor GCF from each group: x(2x + 1) + 3(2x + 1)
Factor out common binomial: (2x + 1)(x + 3)
Verify: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓
What if the trinomial doesn’t factor?
Not every trinomial factors over the integers. If you cannot find two integers with the required product and sum, the polynomial is prime (does not factor). On homework problems, the polynomial will always factor unless the problem specifically asks you to determine whether it is factorable. If you’re stuck, the quadratic formula can confirm whether integer factors exist.
4) Special Patterns
Three special patterns appear frequently enough in algebra that recognizing them on sight is faster than working through a method. All three involve two-term polynomials (binomials) or perfect square trinomials. Once you identify the pattern, the factored form follows directly.
Difference of squares
The pattern a² − b² = (a+b)(a−b) requires two conditions: both terms must be perfect squares, and the sign between them must be minus. If the sign is plus (sum of squares), the polynomial does not factor over real numbers. This is the most commonly misapplied pattern in algebra.
✓ Correct: x² − 16 = (x + 4)(x − 4). The minus sign is required.
Perfect square trinomials
A trinomial is a perfect square when the first and last terms are perfect squares and the middle term equals twice the product of their square roots. When the middle term is positive the factored form is (a+b)²; when negative it is (a−b)².
= (x + 6)²
x² – 14x + 49: first = x², last = 49 = 7², middle = 2(x)(7) = 14x ✓
= (x – 7)²
Sum and difference of cubes
Sum of cubes: a³ + b³ = (a+b)(a² − ab + b²). Difference of cubes: a³ − b³ = (a−b)(a² + ab + b²). The trinomial factor in both cases does not factor further. Recognizing the pattern requires knowing the perfect cubes: 1, 8, 27, 64, 125.
5) Factoring by Grouping
Factoring by grouping is used for four-term polynomials. The process groups the terms into two pairs, factors the GCF out of each pair, and then factors out the common binomial that results.
Group into pairs: (x³ + 3x²) + (2x + 6)
Factor GCF from each pair:
x²(x + 3) + 2(x + 3)
Factor out the common binomial (x + 3):
= (x + 3)(x² + 2)
The method only works cleanly when both pairs produce the same binomial factor. If the binomials don’t match, try rearranging the original four terms into a different grouping before concluding the polynomial is prime.
Group: (3ax – 3ay) + (bx – by)
Factor GCF from each: 3a(x – y) + b(x – y)
Factor out common binomial: (x – y)(3a + b)
When the second pair has a leading negative term, factor out a negative GCF from that pair to make the binomials match.
Group: (x³ – 2x²) + (-3x + 6)
Factor GCF from first pair: x²(x – 2)
Factor GCF from second pair: -3(x – 2) ← factor out -3, not +3
x²(x – 2) – 3(x – 2)
Factor out common binomial: (x – 2)(x² – 3)
Grouping also appears as the final step inside the AC method for trinomials with a ≠ 1, as shown in Section 3 above.
6) Where Students Lose Points
Factoring errors follow predictable patterns. The six below account for the majority of lost points on homework and exams.
Skipping the GCF check
Jumping directly into trinomial factoring or the AC method without first checking for a common factor produces an incomplete result that MyMathLab, ALEKS, and other platforms reject.
✓ Correct: Factor out 2 first: 2(x² + 3x + 2) = 2(x + 1)(x + 2).
Sign errors in trinomial factoring
The signs of p and q in (x + p)(x + q) must satisfy both conditions: p × q = c AND p + q = b. When c is positive and b is negative, both p and q must be negative. When c is negative, one is positive and one is negative.
✓ Correct: Need two negatives: (−3) × (−4) = 12 and (−3) + (−4) = −7. Answer: (x − 3)(x − 4).
Trying to factor a sum of squares
a² + b² does not factor over real numbers. Applying the difference of squares pattern to a sum of squares produces a wrong answer that cannot be verified by multiplication.
✓ Correct: x² + 9 is prime over the real numbers. It does not factor.
Incomplete factoring
Stopping after one level of factoring when the result can be factored further. Every factor in the final answer must be prime — unable to be factored any more.
✓ Correct: (x² + 4)(x + 2)(x − 2). Factor every factorable factor.
AC method sign error when a × c is negative
When a × c is negative, students consistently search only for positive factor pairs and miss the answer. A negative product means one factor must be positive and one negative. The pair that adds to b will have opposite signs.
✓ Correct: a × c = 3 × (−2) = −6. Pairs of −6: (−1, 6), (1, −6), (−2, 3), (2, −3). Which adds to −5? → (1, −6). Rewrite: 3x² + x − 6x − 2 → x(3x + 1) − 2(3x + 1) → (3x + 1)(x − 2).
MyMathLab format issues
MyMathLab requires the factored form to be entered with correct syntax. Multiplication must be entered explicitly where the platform requires it, and parentheses must be placed correctly. ALEKS similarly checks for fully simplified factored form. For platform-specific entry guidance: MyMathLab Help and ALEKS College Algebra Help.
7) How FMMC Can Help
Factoring problems appear on every major algebra platform — MyMathLab, ALEKS, WebAssign, and Hawkes Learning. If factoring homework, quizzes, or exams are holding up your grade, FMMC’s algebra experts handle every assignment with an A/B guarantee.
Algebra Homework
Every factoring module handled accurately and on time — GCF, trinomials, special patterns, grouping, and multi-step problems. See our algebra homework help page.
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Other algebra guides in this series
Factoring connects directly to solving quadratic equations — once a quadratic is factored, setting each factor equal to zero gives the solutions. Also in this series: polynomial functions and graphing lines.
FAQ: Factoring in Algebra
What is factoring in algebra?
Factoring rewrites a polynomial as a product of simpler expressions. It is the reverse of multiplication. For example, x² + 5x + 6 factors to (x + 2)(x + 3). Factoring is used to solve equations, simplify rational expressions, and analyze polynomial functions.
What is the first step in factoring any polynomial?
Always check for a greatest common factor (GCF) first. Factor it out before applying any other method. Skipping this step leads to incomplete factoring, which MyMathLab and ALEKS mark wrong even if the rest of the factoring is correct.
How do you factor a trinomial when the leading coefficient is not 1?
Use the AC method. Multiply the leading coefficient a by the constant c. Find two integers that multiply to that product and add to b. Rewrite the middle term bx using those two integers, then factor by grouping. For example, to factor 2x² + 7x + 3: multiply 2 × 3 = 6, find 1 and 6 (multiply to 6, add to 7), rewrite as 2x² + x + 6x + 3, group and factor to get (2x + 1)(x + 3).
Does x² + 9 factor?
No. A sum of squares (a² + b²) does not factor over real numbers. Only the difference of squares (a² − b²) factors, giving (a+b)(a−b). x² + 9 is prime.
How do you know when a polynomial is fully factored?
A polynomial is fully factored when every factor is either a monomial, a prime polynomial, or a linear binomial that cannot be factored further. After factoring, check each factor individually. If any factor is a difference of squares, a trinomial that factors, or has a common factor, continue factoring. Verify the final answer by multiplying all factors back together — you should recover the original polynomial.
What does it mean when a polynomial is prime?
A prime polynomial cannot be factored into polynomials with integer coefficients. It is the polynomial equivalent of a prime number. Common examples: x² + 9 (sum of squares), x² + x + 1 (no integer pair multiplies to 1 and adds to 1), and any trinomial where no integer factor pair satisfies both the product and sum conditions. When MyMathLab or ALEKS asks you to factor and the answer is prime, enter the original polynomial unchanged or select “prime” if the platform provides that option.
What is the difference between factoring and the quadratic formula?
Factoring rewrites a polynomial as a product; the quadratic formula solves a quadratic equation by finding the values of x that make it equal zero. Both methods solve quadratic equations, but factoring only works when integer factors exist. The quadratic formula works for every quadratic equation including those with irrational or complex solutions. Factoring is faster when it applies; the quadratic formula is the fallback when factoring is not possible.
What is the SOAP method for factoring cubes?
SOAP is a memory device for the signs in the sum and difference of cubes formulas. S = Same sign as the original (the first factor takes the same sign as between the two cubes). O = Opposite sign (the second term in the trinomial factor takes the opposite sign). A = Always positive (the last term in the trinomial is always positive). For a³ + b³: first factor (a+b), trinomial (a² − ab + b²). For a³ − b³: first factor (a−b), trinomial (a² + ab + b²).
Can FMMC help with factoring homework and algebra exams?
Yes. FMMC handles algebra homework, quizzes, and proctored exams across MyMathLab, ALEKS, WebAssign, and Hawkes Learning. See our algebra homework help page or get a free quote.
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