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Home›Algebra›Expressions

Factoring out a common factor

12x + 18 and 6(2x + 3) are the same expression. Only the second shows at a glance that, for any whole number x, the value is a multiple of 6.

Twelve pens, eighteen notebooks, no leftovers

Picture a stationery shop owner with 12 pens and 18 notebooks who wants to make identical gift sets, every set the same and nothing left over. Two sets work: 6 pens and 9 notebooks each. Three sets work too: 4 and 6. The most she can make is 6 sets, each with 2 pens and 3 notebooks, because 6 is the largest number that divides both 12 and 18. In symbols she has turned a sum into a product: 12 + 18 = 6 · 2 + 6 · 3 = 6(2 + 3). Six sets, and each set holds 2 + 3 items. Reading that line from right to left is the distributive property from the last unit. Reading it from left to right is factoring.

Now suppose the order is scaled up: x boxes of 12 pens and 18 loose notebooks. The total is 12x + 18, and the same 6 still divides both parts, so 12x + 18 = 6 · 2x + 6 · 3 = 6(2x + 3). The figure shows it as a rectangle with area 12x + 18. Make it 6 tall and it is 2x + 3 wide. You could also make it 2 tall and 6x + 9 wide, and that is a true statement as well, but 6x + 9 still has a factor of 3 inside it. Taking out the greatest common factor leaves nothing more to take out.

Two rectangles with the same area, 12x + 18, drawn on one scale. Top: height 6, width 2x + 3. Bottom: height 2, width 6x + 9. The bottom one is correct but not finished, because 6x + 9 = 3(2x + 3).

Finding the greatest common factor

Factoring means splitting a product back into its factors. The greatest common factor, or GCF, of two or more expressions is the largest expression that is a factor of all of them. For numbers, list prime factors and keep the ones every number shares. OpenStax’s first example is 54 and 36: 54 = 2 · 3 · 3 · 3 and 36 = 2 · 2 · 3 · 3, so they share 2 · 3 · 3 = 18, and indeed 54 = 18 · 3 and 36 = 18 · 2. Variables work the same way, counting each letter as a factor. 27x³ = 3 · 3 · 3 · x · x · x and 18x⁴ = 2 · 3 · 3 · x · x · x · x share 3 · 3 · x · x · x, so their GCF is 9x³.

  1. Find the GCF of all the terms.
  2. Rewrite each term as the GCF times something.
  3. Use the distributive property in reverse to take the GCF outside a bracket.
  4. Check by multiplying the factors back out.

These are the four steps OpenStax gives, and the last one is the one learners skip. Multiplying back is fast and catches almost every slip. 4x + 12 = 4(x + 3), and 4 · x + 4 · 3 gives back 4x + 12. 12x − 60 = 12(x − 5). 5a + 5 = 5(a + 1), where the 1 matters: without it the bracket would expand to 5a, not 5a + 5. With three terms the GCF must divide all three, as in 4y² + 24y + 28 = 4(y² + 6y + 7). With letters it can include a power, as in 5x³ − 25x² = 5x²(x − 5) and 21x³ − 9x² + 15x = 3x(7x² − 3x + 5).

When the first term is negative, it is usual to take the minus sign out as part of the GCF. OpenStax factors −8y − 24 as −8(y + 3); check: −8 · y = −8y and −8 · 3 = −24. The common factor need not be a single term either. In 5q(q + 7) − 6(q + 7) both parts contain the whole bracket q + 7, so it comes out as one block: (q + 7)(5q − 6). Factoring also explains facts about whole numbers. Three consecutive integers n, n + 1 and n + 2 add up to 3n + 3 = 3(n + 1), which is always a multiple of 3 and always three times the middle number.

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Column B holds 12x + 18. Type =6*(2*A2+3) in C2 and fill it down: every row agrees with B. Try a wrong factoring such as =6*(2*A2+18) to see the check fail, then put it right.
4x + 124(x + 3)
12x − 6012(x − 5)
5a + 55(a + 1)
5x³ − 25x²5x²(x − 5)
−8y − 24−8(y + 3)
21x³ − 9x² + 15x3x(7x² − 3x + 5)
n + (n + 1) + (n + 2)3(n + 1)
  1. Find the GCF of 54 and 36.
  2. Find the GCF of 12 and 18.
  3. Factor 4x + 12.
  4. Factor 12x − 60.
  5. Factor 5a + 5.
  6. Factor −8y − 24, taking the minus sign out.
  7. Factor 5x³ − 25x² completely.
    • 5x²(x − 5)
    • 5x(x² − 5x)
    • 5(x³ − 5x²)
    • x²(5x − 25)
  8. Factor 21x³ − 9x² + 15x completely.
    • 3x(7x² − 3x + 5)
    • 3(7x³ − 3x² + 5x)
    • x(21x² − 9x + 15)
    • 3x(7x² − 9x + 15)
  9. Factor 5q(q + 7) − 6(q + 7).
    • (q + 7)(5q − 6)
    • (q + 7)(5q + 6)
    • 5q − 6(q + 7)
    • (5q − 6)q + 7
  10. Use factoring to work out 37 · 13 + 37 · 7.
  11. The sum of any three consecutive integers is always a multiple of:
    • 3
    • 2
    • 6
    • 9
  12. Evaluate 6(2x + 3) when x = 4.
Common mistakes
  • “6x + 6 = 6x.” Taking 6 out of the second term leaves 1, not nothing: 6x + 6 = 6(x + 1). Expand to check: 6(x) is only 6x.
  • “12x + 18 = 2(6x + 9) is fully factored.” It is true, but 6x + 9 still has a common factor 3. Take out the greatest common factor, 6, to get 6(2x + 3).
  • “−8y − 24 = −8(y − 3).” Expand to check: −8 · (−3) = +24, not −24. Taking out −8 flips both signs inside: −8(y + 3).
Teacher's noteIn the 2022 Korean national curriculum, the greatest common factor appears twice before factorisation: in primary grades 5 to 6 with factors and common factors, and in middle school, where prime factorisation is used to find it. Factorisation itself sits in the middle school standard on multiplying polynomials and factorising, and its explanation asks learners to understand the two as inverse processes. The term list includes factor and factorisation. The curriculum also says learners should see that arithmetic and prime factorisation extend to polynomials, and the gift-set problem makes that link concrete. The step learners skip is the final check by expanding, so make it part of the written answer.
Key points
  • Factoring writes a sum as a product: ab + ac = a(b + c).
  • Find the GCF of every term, numbers and letters, and take it outside the bracket.
  • Check by expanding: 5(a + 1) gives back 5a + 5.
  • Take out the greatest factor, and the minus sign when the first term is negative: −8y − 24 = −8(y + 3).
Sources for this course (10)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수02-01] 다양한 변화 규칙을 찾아 설명하고 수나 식으로 나타내기(규칙을 식으로 나타낼 때 혼합 계산식, 일반항을 나타낸 식 등을 이용해야 하는 복잡한 문제는 다루지 않음); 초등학교 5~6학년군 [6수01-01] 혼합 계산에서 계산하는 순서(계산 순서에 중점, 지나치게 복잡한 혼합 계산은 다루지 않음), [6수01-04] 약수·공약수·최대공약수, [6수02-01] 대응 관계를 □, △ 등을 사용하여 식으로(덧셈식, 뺄셈식, 곱셈식, 나눗셈식 중 하나로 표현되는 간단한 경우만); 중학교 1~3학년군 [9수01-02] 소인수분해로 최대공약수, 수와 연산 용어 거듭제곱·지수·밑·분배법칙, [9수02-01] 다양한 상황을 문자를 사용한 식으로 나타내어 유용성을 인식하고 식의 값 구하기, [9수02-02] 일차식의 덧셈과 뺄셈, [9수02-09] 다항식의 덧셈과 뺄셈, [9수02-10] (단항식)×(다항식), [9수02-14] 함숫값, [9수02-19] 다항식의 곱셈과 인수분해(역관계 이해); 변화와 관계 용어 대입, 다항식, 항, 단항식, 상수항, 계수, 차수, 일차식, 동류항, 등식, 전개, 변수, 인수, 인수분해; '식의 값', '전개식'은 교수·학습 상황에서 사용 가능; 문자와 수, 문자와 일상 언어의 공통점과 차이점 탐색)
  • Textbooks: OpenStax Prealgebra 2e, 2.1 Use the Language of Algebra (Greg 20 and Alex 23, g + 3; variable and constant; × not used in algebra; expression like a phrase, equation like a sentence; 4 + 3·7 gives 25, not 49; PEMDAS and "Please Excuse My Dear Aunt Sally"; multiplication and division left to right, 18 ÷ 9·2; 18 ÷ 6 + 4(5 − 2); 5 + 2³ + 3[6 − 3(4 − 2)]; 2³ + 3⁴ ÷ 3 − 5²; work down the page: openstax.org/books/prealgebra-2e/pages/2-1-use-the-language-of-algebra)
  • 2.2 Evaluate, Simplify, and Translate Expressions (evaluate by substituting; 9x − 2 at x = 5 and x = 1; term, coefficient, coefficient of x is 1; 9b + 15x² + a + 6; like terms 5x, 7, n², 4, 3x, 9n²; 3x + 7 + 4x + 5; 7x² + 8x − x² − 4x; eight more than y, seven less than 9z; window height 6 inches less than width; dimes 2 less than 5 times the quarters: .../pages/2-2-evaluate-simplify-and-translate-expressions)
  • 7.3 Distributive Property (three friends at the movies, 3(9.25) = 3(9 + 0.25) = 27.75; a(b + c) = ab + ac, a(b − c), (b + c)a; 3(x + 4); 2(x − 3); 6(5y + 1); 100(0.3 + 0.25q): .../pages/7-3-distributive-property)
  • 9.4 (perimeter of a rectangle P = 2L + 2W: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
  • OpenStax Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping (GCF of 54 and 36 is 18; GCF of 27x³ and 18x⁴ is 9x³; four steps, distributive property in reverse, check by multiplying; 4x + 12, 5a + 5, 12x − 60, 4y² + 24y + 28, 5x³ − 25x², 21x³ − 9x² + 15x = 3x(7x² − 3x + 5), −8y − 24 = −8(y + 3), 5q(q + 7) − 6(q + 7): openstax.org/books/elementary-algebra-2e/pages/7-1-greatest-common-factor-and-factor-by-grouping)
  • 2.6 (distance, rate and time formula d = rt: .../pages/2-6-solve-a-formula-for-a-specific-variable)
  • Microsoft Support, "Calculation operators and precedence in Excel" (negation before %, ^, then * and /, then + and −; =5+2*3: support.microsoft.com/en-us/office/calculation-operators-and-precedence-in-excel-48be406d-4975-4d31-b2b8-7af9e0e2878a)
  • History for the story block only: MacTutor History of Mathematics, University of St Andrews, "François Viète" (1540 to 1603; studied law, son of a lawyer; decoded letters to Philip II of Spain for Henry IV; In artem analyticam isagoge, Tours 1591, first systematic algebraic notation, vowels for unknowns and consonants for knowns; convention of early letters for knowns and late letters for unknowns introduced later by Descartes in La Géométrie: mathshistory.st-andrews.ac.uk/Biographies/Viete/)
  • "René Descartes" (1596 to 1650; La Géométrie an appendix to the Discours de la méthode, published at Leiden in 1637: mathshistory.st-andrews.ac.uk/Biographies/Descartes/).

Checked: 2026-10-19