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➕ Algebra 1 · Factoring

Factoring Out the Greatest Common Factor

Factor out the greatest common factor of numbers and terms with variables, check by distributing, handle negatives, and know when an expression is completely factored.

Grades 9, 10⏱ 14 min✔ AI-checkedCCSS A-SSE.A.2CCSS A-SSE.B.3CCSS 6.EE.A.3CCSS 6.NS.B.4
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
  • A-SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • 6.EE.A.3: Apply the properties of operations to generate equivalent expressions.
  • 6.NS.B.4: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.

By the end you can…

  • Find the greatest common factor (GCF) of numbers, and of terms with variables by using the lowest exponent of each variable that every term shares.
  • Factor the GCF out of a polynomial and check the result by distributing.
  • Factor out a negative GCF when the leading coefficient is negative, and recognize when the GCF is 1 and there is nothing to pull out.
  • Pull out a common binomial in simple cases, and tell when an expression is completely factored.

Lesson (school style)

Factoring is multiplying backward: you rewrite a sum like 6x + 9 as a product like 3(2x + 3). The simplest kind of factoring pulls out the greatest common factor (GCF), the largest piece that divides every term evenly. Make it your first step. It is the distributive property read from right to left, and you can check it by distributing.

Factoring is multiplying backward

When you multiply, you turn a product into a sum: 5(3x + 7) = 15x + 35. Factoring goes the other way. You start with 15x + 35 and look for a product that equals it. This works because both terms contain the same number: 15x = 5 · 3x and 35 = 5 · 7, so 15x + 35 = 5(3x + 7).

That is the distributive property run backward: ab + ac = a(b + c). The shared piece a is a common factor, and you pull it out in front of the parentheses.

A factor of a number divides it evenly (the whole-number factors of 12 are 1, 2, 3, 4, 6 and 12). You can check any factoring the same way: multiply the answer back out and see whether you get the original expression.

The GCF of numbers

The greatest common factor of two or more numbers is the largest number that divides every one of them evenly. It is always positive.

Method 1: list the factors. For 24 and 36:

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  • The numbers in both lists are 1, 2, 3, 4, 6 and 12, so the GCF is 12.

Method 2: prime factors. Write each number as a product of primes: 24 = 2 · 2 · 2 · 3 and 36 = 2 · 2 · 3 · 3. Keep the primes the numbers share, using each prime the fewest times it appears in any one of the numbers. Here that is 2 · 2 · 3 = 12. (24 has three 2’s but 36 has only two, so only two 2’s are shared.)

Two quick cases: if one number divides the other, it is the GCF (the GCF of 7 and 21 is 7). If the numbers share nothing but 1, the GCF is 1.

The GCF of terms with variables

For terms like 12a3b2 and 18a2b, build the GCF one piece at a time:

  1. Number part: the GCF of the coefficients.
  2. Each variable: include a variable only if it appears in every term, and use its lowest exponent. (A letter with no visible exponent has exponent 1.)

For 12a3b2 and 18a2b: the GCF of 12 and 18 is 6. The letter a appears in both, with exponents 3 and 2, so use a2. The letter b appears in both, with exponents 2 and 1, so use b. The GCF is 6a2b.

Why the lowest exponent? By the laws of exponents, a3 = a2 · a, so a2 fits inside a3 and inside a2. But a3 does not fit inside a2: dividing a2 by a3 leaves a−1, which is not a polynomial term. So the most you can take out of every term is the lowest power.

A letter that is missing from even one term cannot be in the GCF. For 14m2n and 21m3, the GCF is 7m2 (the second term has no n). A constant term has no letters, so 5x2 + 10 has GCF 5 and no x.

Factoring it out: divide each term

To factor out the GCF:

  1. Find the GCF of all the terms.
  2. Write the GCF, then parentheses.
  3. Divide each term by the GCF, and write the quotients inside, keeping each sign.
  4. Check by distributing.

To divide terms with letters, divide the coefficients and subtract the exponents: 12x3 ÷ (4x2) = 3x. (Assume any variable you divide by is not zero. The finished factorization is true for every value of the variable, including 0, as the check by distributing shows.)

The parentheses hold as many terms as the original expression. If one term is exactly the GCF, its quotient is 1, not nothing: 9z2 + 3z has GCF 3z, and 3z ÷ (3z) = 1, so 9z2 + 3z = 3z(3z + 1).

Check by distributing

Multiply the GCF back into every term inside the parentheses. You should land exactly on the original expression. For 3z(3z + 1): 3z · 3z = 9z2 and 3z · 1 = 3z, which matches. If it does not match, look for a wrong sign or a missing 1.

Distributing shows that your factored form equals the original. It does not show that you took out the greatest common factor. That takes a second look inside the parentheses (see “Completely factored” below).

A negative leading coefficient

When a polynomial is written with its highest power first (standard form), its leading coefficient is the number on the first term. If that number is negative, it is usual to factor out the negative of the GCF, so that the first term inside the parentheses is positive.

For −8x2 + 20x the GCF is 4x. Factor out −4x: −8x2 + 20x = −4x(2x − 5).

Dividing by a negative number flips the sign of every term: −8x2 ÷ (−4x) = 2x and 20x ÷ (−4x) = −5.

Both −4x(2x − 5) and 4x(−2x + 5) multiply out to the original, so both are correct. Factoring out the negative is a convention: it keeps the first term inside positive, which makes later factoring easier. (The GCF itself is always positive. Here you choose to pull out its negative.)

When the GCF is 1

Sometimes the only common factor is 1. In 9x + 14, the numbers 9 and 14 share only the factor 1, and 14 has no x. The GCF is 1, so there is nothing to pull out. (Writing 1(9x + 14) is true but changes nothing.)

This lesson’s method finds only common factors. A GCF of 1 does not mean an expression can never be factored: x2 + 7x + 10 has GCF 1, yet it equals (x + 2)(x + 5). You will learn that kind of factoring in the next lesson, Factoring Trinomials.

A common binomial factor

The common factor can have more than one term. In x(x + 4) + 6(x + 4), both terms contain the binomial (x + 4). Treat it as one chunk and pull it out. What is left from each term goes in the other parentheses, keeping signs: x and 6.

x(x + 4) + 6(x + 4) = (x + 4)(x + 6).

If a term is just the binomial, its leftover is 1: 7(y − 1) + (y − 1) = (y − 1)(7 + 1) = 8(y − 1).

This simple case works when the same binomial, written the same way, appears in every term. (Opposite binomials such as (x − 5) and (5 − x) need one more step, which you will meet in factoring by grouping.) As always, check by expanding: (x + 4)(x + 6) = x2 + 10x + 24, and x(x + 4) + 6(x + 4) = x2 + 4x + 6x + 24 = x2 + 10x + 24.

Completely factored

An expression is completely factored when it is written as a product of factors that cannot be factored any further using integers. For this lesson, the key test is: after you pull out the GCF, no common factor greater than 1 remains inside the parentheses.

If you pull out only part of the GCF, the work is correct but unfinished. 28x + 42 equals 7(4x + 6), but 4 and 6 still share the factor 2, so keep going: 14(2x + 3). Starting with the full GCF, 14, saves a step.

Two reminders. First, distributing to check shows your answer is correct, not that it is complete. Second, later lessons add more ways to factor, and an expression is only completely factored when none of them applies. For example, 2x(x2 − 9) is not finished, because x2 − 9 = (x − 3)(x + 3). In this lesson’s worked examples and practice questions the leftover parentheses do not factor any further, so “no common factor left” is enough there.

A number or letter group out front, such as 14 or 6x, counts as one factor. You do not need to break it into 2 · 7 or 2 · 3 · x.

Why factor?

Factored form shows structure that the expanded form hides.

  • Divisibility. 15x + 35 = 5(3x + 7) shows that for any whole number x, the value is a multiple of 5. Try x = 4: 15(4) + 35 = 95, and 5(3 · 4 + 7) = 5 · 19 = 95.
  • Dimensions. If a rectangle has area 10x2 + 6x, then 10x2 + 6x = 2x(5x + 3) shows it could be 2x by (5x + 3).
  • Solving equations. A product of numbers is 0 only when at least one factor is 0. So x(x + 6) = 0 means x = 0 or x = −6. That is why factoring will matter for solving equations like x2 + 6x = 0 later in the course.

Key words

Factor (noun)
A number or expression that is multiplied by another to make a product. 3 and 4 are factors of 12, and x and (x + 2) are factors of x2 + 2x.
Factoring
Rewriting a sum as a product, such as 15x + 35 = 5(3x + 7). It is multiplying run backward.
Greatest common factor (GCF)
The largest expression that divides every term evenly: the GCF of the coefficients, times each variable that appears in every term at its lowest exponent. For 6x3 + 10x it is 2x.
Monomial
A number, a variable, or a product of numbers and variables with whole-number exponents, such as 6, x or −4x2.
Polynomial
A sum of monomials, called terms. A polynomial with two terms is a binomial, and one with three terms is a trinomial.
Coefficient
The number multiplied by the variable part of a term, with its sign. In −8x2 the coefficient is −8.
Leading coefficient
The coefficient of the term with the highest power, when the polynomial is written in standard form (descending powers). In −8x2 + 20x it is −8.
Common binomial factor
A binomial, such as (x + 4), that appears as a factor in every term of an expression.
Completely factored
Written as a product of factors that cannot be factored further using integers. After pulling out a GCF, one necessary check is that the terms inside the parentheses share no common factor greater than 1. A number out front, such as 14, counts as one factor.

Worked examples

Example 1. Find the GCF of 24 and 36.

  1. List the factors of each number. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  2. The common factors (in both lists) are 1, 2, 3, 4, 6 and 12. The greatest is 12.
  3. Check with prime factors: 24 = 2 · 2 · 2 · 3 and 36 = 2 · 2 · 3 · 3. Both contain 2 · 2 · 3 (36 has only two 2’s, so only two are shared), and 2 · 2 · 3 = 12.
  4. Check that nothing larger works: 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and 2 and 3 share no factor larger than 1.

Answer: Answer: The GCF of 24 and 36 is 12.

Example 2. Find the GCF of (a) 12a3b2, 18a2b and 30a5b4, and (b) 14m2n and 21m3.

  1. (a) Coefficients: 12 = 6 · 2, 18 = 6 · 3 and 30 = 6 · 5, and 2, 3 and 5 share no factor larger than 1, so the number part is 6.
  2. (a) The letter a is in every term, with exponents 3, 2 and 5. Use the lowest, 2: a2.
  3. (a) The letter b is in every term, with exponents 2, 1 and 4. Use the lowest, 1: b. The GCF is 6a2b.
  4. (a) Check: 12a3b2 ÷ (6a2b) = 2ab, 18a2b ÷ (6a2b) = 3 and 30a5b4 ÷ (6a2b) = 5a3b3. Each division works, and the quotients 2ab, 3 and 5a3b3 share no common factor greater than 1.
  5. (b) The GCF of 14 and 21 is 7. The letter m has exponents 2 and 3, so use m2. The letter n is missing from 21m3, so it is not in the GCF.
  6. (b) The GCF is 7m2. Check: 14m2n ÷ (7m2) = 2n and 21m3 ÷ (7m2) = 3m.

Answer: Answer: (a) 6a2b. (b) 7m2.

Example 3. Factor 15x + 35.

  1. The GCF of 15 and 35 is 5. The constant 35 has no letter, so x is not in the GCF. The GCF is 5.
  2. Divide each term by 5: 15x ÷ 5 = 3x and 35 ÷ 5 = 7.
  3. Write the GCF outside and the quotients inside, keeping the plus sign: 5(3x + 7).
  4. Check by distributing: 5 · 3x = 15x and 5 · 7 = 35, so 5(3x + 7) = 15x + 35.
  5. Completely factored? Inside, 3 and 7 share no factor greater than 1, and 7 has no x. Yes.

Answer: Answer: 5(3x + 7)

Example 4. Factor 6x3 − 9x2 + 12x.

  1. The GCF of 6, 9 and 12 is 3.
  2. The letter x is in all three terms, with exponents 3, 2 and 1. The lowest is 1, so the GCF is 3x.
  3. Divide each term by 3x (assume x ≠ 0 while dividing): 6x3 ÷ (3x) = 2x2, −9x2 ÷ (3x) = −3x and 12x ÷ (3x) = 4.
  4. Write the quotients inside, keeping their signs: 3x(2x2 − 3x + 4). The original has three terms and so does the inside.
  5. Check by distributing: 3x · 2x2 = 6x3, 3x · (−3x) = −9x2 and 3x · 4 = 12x. It matches the original.
  6. Test with x = 2: the original is 48 − 36 + 24 = 36 and the factored form is 3(2)(8 − 6 + 4) = 6 · 6 = 36. Inside, the coefficients 2, −3 and 4 share only 1, and the last term 4 has no x, so no common factor is left. It is complete.

Answer: Answer: 3x(2x2 − 3x + 4)

Example 5. Factor 12x3 + 4x2.

  1. The GCF of 12 and 4 is 4. The letter x has exponents 3 and 2, so use x2. The GCF is 4x2.
  2. Divide each term by 4x2 (assume x ≠ 0 while dividing): 12x3 ÷ (4x2) = 3x and 4x2 ÷ (4x2) = 1.
  3. The second term equals the GCF, so it leaves a 1, not nothing: 4x2(3x + 1).
  4. Check by distributing: 4x2 · 3x = 12x3 and 4x2 · 1 = 4x2. Leaving out the 1 would give 4x2(3x) = 12x3, which has lost the 4x2 term.

Answer: Answer: 4x2(3x + 1)

Example 6. Factor −8x2 + 20x by taking out the negative of the GCF, so the first term inside the parentheses is positive.

  1. The GCF of 8 and 20 is 4, and the lowest exponent of x is 1, so the GCF is 4x. The leading coefficient is negative, so factor out −4x instead.
  2. Divide each term by −4x: −8x2 ÷ (−4x) = 2x, because a negative divided by a negative is positive. And 20x ÷ (−4x) = −5.
  3. Write the quotients inside: −4x(2x − 5).
  4. Check by distributing: −4x · 2x = −8x2 and −4x · (−5) = +20x. It matches.
  5. The form 4x(−2x + 5) also equals the original. We prefer −4x(2x − 5) because it keeps the first term inside positive.

Answer: Answer: −4x(2x − 5)

Example 7. Factor 28x + 42 completely.

  1. Suppose you spot only that 28 and 42 are both divisible by 7. Pulling out 7 gives 7(4x + 6). Check: 7 · 4x = 28x and 7 · 6 = 42, so this is correct.
  2. But is it complete? Inside, 4 and 6 are both divisible by 2, so a common factor remains.
  3. Pull out the 2: 4x + 6 = 2(2x + 3), so 7(4x + 6) = 7 · 2(2x + 3) = 14(2x + 3).
  4. Starting with the full GCF would have saved a step: the GCF of 28 and 42 is 14.
  5. Now inside, 2x + 3 has coefficients 2 and 3, which share only 1. No common factor remains.

Answer: Answer: 14(2x + 3)

Example 8. Factor 9x + 14 if you can pull out a common factor.

  1. The factors of 9 are 1, 3 and 9. The factors of 14 are 1, 2, 7 and 14. The only common factor is 1.
  2. The 14 has no x, so no letter is common. The GCF is 1.
  3. Writing 1(9x + 14) is true but changes nothing, so there is nothing to pull out. The expression stays 9x + 14.
  4. A GCF of 1 only says there is no common factor to remove. Some expressions with GCF 1, such as x2 + 7x + 10, can still be factored in other ways (next lesson).

Answer: Answer: 9x + 14 has no common factor greater than 1 to pull out (its GCF is 1).

Example 9. Factor 2x(x + 5) − 3(x + 5).

  1. Both terms contain the binomial (x + 5): the first is 2x times it, and the second is −3 times it.
  2. Treat (x + 5) as one chunk and pull it out. What is left from the terms is 2x and −3, with the minus sign staying with the 3: (x + 5)(2x − 3).
  3. Check by expanding: (x + 5)(2x − 3) = 2x2 − 3x + 10x − 15 = 2x2 + 7x − 15. The original expands to 2x2 + 10x − 3x − 15 = 2x2 + 7x − 15. They match.
  4. Test with x = 4: the original is 2 · 4 · 9 − 3 · 9 = 72 − 27 = 45, and the factored form is 9 · 5 = 45.

Answer: Answer: (x + 5)(2x − 3)

In plain English

Factoring is multiplying backward. Say a teacher has 16 markers and 24 sticky-note pads. She wants identical kits with nothing left over.

Both 16 and 24 can be divided by 8. So she can make 8 kits. Each kit has 2 markers and 3 pads. In math, 16 + 24 = 8(2 + 3).

Letters work the same way. In 6x + 9, both parts can be divided by 3. So you pull the 3 out front and write 3(2x + 3). The 3 is the greatest common factor, or GCF. It is the biggest piece that goes into every part.

When the parts share a letter, use the smaller power. The parts x3 and x2 share x2, because x2 fits inside both. Divide each part by the GCF and put what is left inside the parentheses.

Always check. Multiply back out. If you get the first expression, you are right. Then look inside the parentheses. If every part still shares something, pull that out too.

See it

The GCF is what the prime factors have in common 24 = 2 · 2 · 2 · 3 36 = 2 · 2 · 3 · 3 2 3 2 2 3 only in 24 only in 36 shared GCF = 2 × 2 × 3 = 12 What is left of each number goes inside: 24 + 36 = 12(2 + 3) A prime counts only as many times as both numbers have it.
The overlap of the prime factors of 24 and 36 is 2 · 2 · 3, so the GCF is 12. What is left of each number, 2 and 3, goes inside the parentheses: 24 + 36 = 12(2 + 3).
Divide each term by the GCF; the quotients go inside GCF of the coefficients 6, 9, 12 is 3. The lowest exponent of x is 1. So the GCF is 3x 6x3 −9x2 12x ÷ 3x÷ 3x÷ 3x 2x2 −3x 4 Each quotient keeps its sign, because we divided by the positive GCF. 6x3 − 9x2 + 12x = 3x(2x2 − 3x + 4) Check: 3x · 2x2 = 6x3, and 3x · (−3x) = −9x2, and 3x · 4 = 12x
Find the GCF, divide each term by it, and write the quotients inside the parentheses. Then distribute to check.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Relationships

Mutual friends

Compare your friend list with a classmate’s. The friends you both have are your mutual friends. Now treat primes as friends. Write 20 = 2 · 2 · 5 and 50 = 2 · 5 · 5. The mutual friends of the two numbers are one 2 and one 5. The number 20 has two 2’s but 50 has only one, so only one 2 is shared. The number 50 has two 5’s but 20 has only one, so only one 5 is shared.

How it connects: The GCF is the product of the shared primes: 2 · 5 = 10, so the GCF of 20 and 50 is 10. This is the lowest-count rule: each prime (and each variable) is used as many times as the number that has the fewest copies. The limit: a GCF is a product of the shared factors, not a list, and a friend can only be shared as many times as both lists have it.

Food

Fruit cups

You are making identical fruit cups, with nothing left over, from 32 strawberries and 48 blueberries. The most cups you can make is 16, because 16 is the largest number that divides both 32 and 48. Each cup gets 2 strawberries and 3 blueberries.

How it connects: 32 + 48 = 16(2 + 3). The GCF, 16, is the number of cups, and the parentheses hold what is in one cup. Fewer cups also work. Eight cups of 4 strawberries and 6 blueberries gives 8(4 + 6), but that is not completely factored, because each of those cups could still be split into two identical smaller cups. The limit: counting fruit uses whole numbers only. With letters, one “cup” can hold variable amounts, like 4x2.

Proverb

A chain is only as strong as its weakest link

This proverb means that something is only as strong as its weakest part. Think of 10x9, 15x6 and 5x2 as stacks of x’s: 9 of them, 6 of them and 2 of them. If you want to take the same number of x’s out of every stack, you can take at most 2, the size of the shortest stack. The weakest link sets the limit.

How it connects: The GCF uses the lowest exponent: x2 is the most you can pull out of all three terms, so 10x9 + 15x6 + 5x2 = 5x2(2x7 + 3x4 + 1). The limit: the weakest-link rule is only for the exponents of the letters. For the numbers, use the greatest common factor of the coefficients, which is not always the smallest coefficient. The GCF of 6, 4 and 8 is 2, not 4.

Idiom

Find common ground

To “find common ground” means to discover what two sides already agree on. Say one friend wants noodles with chicken and another wants noodles with tofu. The common ground is noodles. Set that aside and only the topping is left to decide.

How it connects: In 6x2 + 9x, the common ground is 3x, because both terms already contain it. Pull it out and only what differs is left inside: 3x(2x + 3). The limit: common ground has to be shared by every term. In 6x2 + 9 the letter x is missing from the second term, so x is not common ground, and only the 3 can be pulled out: 3(2x2 + 3).

Sports

Instant replay

In many sports, officials can review video to check a call. The replay does not change the play. It only checks whether the call matched what happened.

How it connects: Distributing your answer back out is your instant replay. If 5y2(2y + 3) multiplies out to 10y3 + 15y2, the original, the factoring is correct. The limit: a replay confirms only that your answer equals the original. It cannot tell you whether you took out the greatest common factor. 3(4x + 6) replays correctly to 12x + 18, yet it is not completely factored, because 4x + 6 still has the common factor 2: 6(2x + 3).

Everyday joke

The promoted 3

Why did the 3 get promoted to the front office? In 6x + 9 it was the one thing both terms had in common, so it got pulled out front: 3(2x + 3).

How it connects: Pulling the 3 out front does not change the value: 3 · 2x + 3 · 3 = 6x + 9. The limit: only a factor that every term has can move up front. In 6x + 10 the 3 does not divide 10, so the 3 stays put. (The GCF there is 2.)

Memory help

Acronym

PULL it out

Pick the biggest number that divides every coefficient. Use each variable that appears in every term, at its lowest exponent. Leave the leftovers inside: divide each term by the GCF. Look back: distribute to check, then look inside for anything still common. Say it as “PULL it out front.”

Rhyme

The GCF rhyme

Biggest number they all share, lowest power of letters found in every term there. Pull it out and leave the rest; multiply back to pass the test.

Picture it

The overlap picture

Draw two overlapping circles, one for each number. Write each number’s prime factors in its circle, and put the primes both numbers share in the overlap. Multiply what is in the overlap to get the GCF. For 24 and 36 the overlap holds 2, 2, 3, and 2 · 2 · 3 = 12. (If a prime appears three times in one number and twice in the other, only two copies go in the overlap.)

Sketch it

The three-column table

Make a table with the columns Term, ÷ GCF, and Leftover. Write each original term in the first column and its quotient in the third. The leftovers, with their signs and in the same order, go inside the parentheses. A row that leaves 1 shows you the 1 you must not forget.

Trap alert

The vanishing 1

5m3 + 5m2 has GCF 5m2, and the second term gives 5m2 ÷ (5m2) = 1. So the answer is 5m2(m + 1), not 5m2(m). Distribute to see why: 5m2 · m is only 5m3, and the second term is gone. If a term equals the GCF, a 1 stays behind.

Trap alert

Lowest power, not highest

4x2 + 6x5 has GCF 2x2, not 2x5. Only x2 fits inside both terms, since x5 does not fit inside x2. And a letter missing from any term is out: 6a2 + 9b has GCF 3 and nothing else.

Study notes

Key ideas

  • Factoring is multiplying backward. Pulling out a common factor is the distributive property run backward: ab + ac = a(b + c).
  • The GCF of terms is the GCF of the coefficients, times each variable that appears in every term, raised to the lowest exponent found in the terms.
  • To factor it out, divide each term by the GCF and write the quotients inside the parentheses with their signs. A term equal to the GCF leaves a 1.
  • Check by distributing. That confirms the answer is correct, but not that it is complete.
  • If the leading coefficient is negative, factor out the negative of the GCF. If the GCF is 1, there is nothing to pull out. A common factor can also be a whole binomial.
  • An expression is completely factored when no common factor greater than 1 remains inside the parentheses (and, in later lessons, when no other method applies either).

Steps

  1. Write the polynomial in standard form (descending powers) so you can see the leading coefficient.
  2. Find the GCF of the coefficients.
  3. Find each variable that appears in every term, and use its lowest exponent.
  4. Multiply the pieces to get the GCF. If the leading coefficient is negative, use the negative of the GCF.
  5. Divide each term by the GCF and write the quotients inside parentheses, keeping signs.
  6. Check by distributing, then look inside the parentheses for any common factor that is left.

Formulas

  • Factoring (distributive property backward): ab + ac = a(b + c)
  • Factoring out a negative: −ab + ac = −a(b − c)
  • Lowest power: xm and xn with m ≤ n have GCF xm, because xn = xm · xn−m
  • Common binomial factor: a(b + c) + d(b + c) = (b + c)(a + d)

Watch out

  • Use the lowest exponent, not the highest: the GCF of 4x2 and 6x5 includes x2, not x5.
  • A letter must appear in every term to be in the GCF. A constant term has no letters.
  • Do not drop the 1. If a term equals the GCF, its quotient is 1: 3z + 9z2 = 3z(1 + 3z).
  • Dividing by a negative flips every sign: −10t + 25 = −5(2t − 5).
  • Distributing checks that your answer is correct. It does not check that you took out the greatest factor, so look inside the parentheses too.

Why it works

  • Why does it work? a(b + c) means ab + ac, so reading it backward lets you pull a shared a out of both terms.
  • Why the lowest exponent? x5 = x2 · x3, so x2 fits inside x5. But x2 ÷ (x5) would leave x−3, which is not a polynomial term, so x5 cannot be pulled out of x2.
  • Why bother? Factored form shows that a value is a multiple of the GCF, gives the dimensions of a rectangle from its area, and prepares you to solve equations, because a product is 0 only when one of its factors is 0.

Common mistakes

✗ MistakeThe GCF of x5 and x2 is x5 (use the biggest exponent).
✓ InsteadUse the lowest exponent. x2 fits inside both x5 and x2, but x5 does not fit inside x2. The GCF is x2.
✗ Mistake3a2 + 3a = 3a(a) (nothing is left of the second term).
✓ Instead3a ÷ (3a) = 1, so a 1 is left: 3a2 + 3a = 3a(a + 1). Distribute 3a(a) and you only get 3a2.
✗ Mistake−10t + 25 = −5(2t + 5) (forgetting that dividing by a negative flips the sign).
✓ Instead25 ÷ (−5) = −5, so −10t + 25 = −5(2t − 5). Check: −5 · 2t = −10t and −5 · (−5) = +25.
✗ Mistake10x + 20 = 5(2x + 4) is completely factored, because it multiplies out correctly.
✓ InsteadMultiplying out only shows the answer is correct. Inside, 2x + 4 still has the common factor 2, so the complete factoring is 10(x + 2).
✗ MistakeIf the GCF is 1, the expression cannot be factored at all.
✓ InsteadA GCF of 1 only means there is no common factor to pull out. Other methods can still work, such as x2 + 7x + 10 = (x + 2)(x + 5) in the next lesson.
✗ MistakeA letter that appears in most of the terms can go in the GCF.
✓ InsteadA letter must appear in every term. In 6a2 + 9b, neither letter is in both terms, so the GCF is 3.

Teach it back

Explain to a friend how to factor −12x3 + 18x2 so that the first term inside the parentheses is positive, and how you would check your answer.

Explaining it out loud shows what you really know. Did you…

  • I found the GCF of 12 and 18 (which is 6) and the lowest exponent of x (which is 2), so the GCF is 6x2.
  • Because the leading coefficient is negative, I factored out −6x2 and divided each term by it: −12x3 ÷ (−6x2) = 2x and 18x2 ÷ (−6x2) = −3.
  • I wrote −6x2(2x − 3) and checked by distributing: −6x2 · 2x = −12x3 and −6x2 · (−3) = 18x2.
  • I looked inside the parentheses and saw that 2 and 3 share no common factor, so it is completely factored.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Practice questions

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The Distributive PropertyLaws of ExponentsFactoring TrinomialsFactoring by GroupingDifference of SquaresMultiplying Polynomials