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Home›Algebra›Expressions

The distributive property and expanding brackets

7 · 98 is awkward in your head until you write it as 7 · (100 − 2) = 700 − 14 = 686.

Try this before reading on: 7 · 98

Work out 7 · 98 without paper. Most people who manage it quickly do not multiply 98 by 7 digit by digit. They notice that 98 is 2 short of 100, take seven hundreds, which is 700, and remove seven twos, which is 14. The answer is 686. OpenStax gives a money version of the same trick. Three friends going to the movies each need $9.25, that is 9 dollars and 1 quarter. Three times 9 dollars is $27, three quarters are 75 cents, and the total is $27.75. In both cases one multiplication was split into two easier ones and the results added or subtracted.

What you just used has a name, the distributive property. For any numbers a, b and c, a(b + c) = ab + ac. The multiplier outside the bracket is shared out, or distributed, to each term inside. The movie money reads 3(9 + 0.25) = 3 · 9 + 3 · 0.25 = 27 + 0.75 = 27.75, and the mental trick reads 7(100 − 2) = 700 − 14. OpenStax lists two other forms that follow from the first: a(b − c) = ab − ac, and (b + c)a = ba + ca, since the order of a product does not matter.

The rectangle that proves it

Why should the rule hold for every number, not only for the two we tried? Draw it. A rectangle 3 units tall and x + 4 units wide has area 3(x + 4). Cut it with one vertical line where the x part ends. The left piece is 3 by x, area 3x, and the right piece is 3 by 4, area 12. Cutting a rectangle does not change how much space it covers, so 3(x + 4) = 3x + 12. The top picture below shows this cut. The argument works for any width x, which is exactly what a rule about expressions needs.

Top: a 3 by (x + 4) rectangle cut into 3x and 12. Bottom: an (x + 2) by (x + 3) rectangle cut into four pieces, x², 3x, 2x and 6, which add to x² + 5x + 6.

Expanding with letters and minus signs

In algebra the distributive property removes brackets that the order of operations cannot open. The rule says to work inside the bracket first, but in 3(x + 4) there is nothing to do there, since x and 4 are not like terms. So distribute: 3 · x + 3 · 4 = 3x + 12. Writing an expression without its brackets this way is called expanding. More examples from OpenStax: 2(x − 3) = 2x − 6, 6(5y + 1) = 30y + 6, and 100(0.3 + 0.25q) = 30 + 25q. The multiplier must reach every term. Some students draw a little arrow from the outside number to each term inside as a reminder.

Negative multipliers carry their sign to every term. −2(x − 5) is (−2) · x + (−2) · (−5) = −2x + 10. A bare minus in front of a bracket means −1 times the bracket, so −(x + 3) = −x − 3 and −(2x − 4) = −2x + 4, which is the rule the previous unit used without proof. After expanding, collect like terms as usual: 4(x + 2) + 3x = 4x + 8 + 3x = 7x + 8. The graph lab below shows what happens if the multiplier misses a term. It draws y = 2(x + 3), expanded correctly as 2x + 6, beside the common slip 2x + 3.

Loading

Drag to turn. Scroll or pinch to zoom.

Line 1 is 2(x + 3) expanded correctly, 2x + 6. Line 2 is the slip 2x + 3, where the 2 never reached the 3. The two lines are parallel, so the expressions never take the same value. Drag intercept 2 up to 6 and the lines merge: only then are the expressions equal for every x.

Two brackets: every term meets every term

The same idea handles a product of two brackets. In (x + 2)(x + 3), treat x + 2 as the multiplier and distribute it over x + 3: (x + 2)x + (x + 2)3. Distribute again: x² + 2x + 3x + 6, and collect: x² + 5x + 6. The lower rectangle in the figure shows the four pieces, one for each pairing of a term from the first bracket with a term from the second. A quick check: at x = 1 the brackets give 3 · 4 = 12 and the expansion gives 1 + 5 + 6 = 12. Two brackets with two terms each always make four products before collecting, so count them if you are unsure.

  1. Work out 7 · 98 using 7(100 − 2).
  2. Three tickets at $9.25 each. Use 3(9 + 0.25) to find the total.
  3. Expand 3(x + 4).
  4. Expand 2(x − 3).
  5. Expand 6(5y + 1).
  6. Expand −2(x − 5).
  7. Expand −(x + 3).
  8. Expand and simplify 4(x + 2) + 3x.
  9. Expand 100(0.3 + 0.25q).
  10. Which is the expansion of (x + 2)(x + 3)?
    • x² + 5x + 6
    • x² + 6
    • x² + 6x + 5
    • 2x + 5
  11. Evaluate 3(x + 4) when x = 2.
  12. Which is equal to 5(a − 4)?
    • 5a − 20
    • 5a − 4
    • a − 20
    • 5a + 20
Common mistakes
  • “3(x + 4) = 3x + 4.” The 3 has to reach the 4 as well. At x = 0, 3(x + 4) is 12 and 3x + 4 is 4. The graph lab shows the two as parallel lines that never meet.
  • “−2(x − 5) = −2x − 10.” The second product is (−2) · (−5), which is positive: −2x + 10. Check at x = 5: the bracket is 0, so the value must be 0, and −10 − 10 is not 0.
  • “(x + 3)² = x² + 9.” (x + 3)² is (x + 3)(x + 3), which gives four products: x² + 3x + 3x + 9 = x² + 6x + 9. At x = 1 the bracket gives 16 and x² + 9 gives 10.
Teacher's noteThe 2022 Korean national curriculum lists the distributive property among the middle school number terms, where it first serves calculations with integers and rational numbers. It returns in the algebra strand with the multiplication of a monomial by a polynomial, and later with the multiplication of polynomials and factorisation, where expanding is a listed term. Learners already know rectangle areas from primary school, so the area model costs almost no new teaching. A common gap is that learners can expand 3(x + 4) but cannot explain why 3(x + 4) is not 3x + 4. Have them evaluate both at x = 0; the difference of 8 shows up at once.
Key points
  • Distributive property: a(b + c) = ab + ac, and a(b − c) = ab − ac.
  • The multiplier reaches every term inside, with its sign: −2(x − 5) = −2x + 10.
  • A bare minus before a bracket is −1 times it: −(x + 3) = −x − 3.
  • Two brackets give four products: (x + 2)(x + 3) = x² + 5x + 6.
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