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Home›Algebra›Expressions

Terms, coefficients and like terms

Put any number in for x and 3x + 5x comes out equal to 8x. At x = 7 that is 21 + 35 = 56, and 8 · 7 = 56.

Test two expressions before you trust a rule

Here is a claim to test rather than accept: 3x + 5x is the same expression as 8x. “The same” has a precise meaning. Two expressions are equivalent when they give the same value for every number you put in. Testing every number is impossible, but testing several is quick, and one failure is enough to sink a claim. The sheet below already holds five values of x, the expression 3x + 5x, and a rival guess, 8x², that some learners write because they think adding two x terms should give x squared. Your job is column C.

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Columns B and D are filled in. Write =8*A2 in C2 and fill it down. Column C matches column B in every row. Column D matches only at x = 1 and would also match at x = 0, which is why one lucky value proves nothing.

The table makes the point with numbers, and the reason behind it is just as plain in words. 3x means three lots of x and 5x means five lots of x, so together they are eight lots of x, whatever x is. OpenStax uses oranges: 3 oranges and 6 more oranges make 9 oranges. Squaring never happens, because nothing was multiplied by x a second time. At x = 1 the rival 8x² happens to agree, since 1² = 1. At x = 2 it gives 32 instead of 16, and at x = 7 it gives 392 instead of 56. One mismatch is enough.

Naming the parts: terms and coefficients

An expression is built from terms. OpenStax defines a term as a constant or the product of a constant and one or more variables, so 7, y, 5x², 9a and 13xy are all terms. The number that multiplies the variables is the coefficient. In 9b + 15x² + a + 6 the terms are 9b, 15x², a and 6, and the coefficients of the variable terms are 9, 15 and 1. The coefficient of a is 1 even though nobody writes it, because a = 1 · a. A term with no variable, such as 6, is a constant term. When an expression has a minus sign, read it as adding a negative: 5x − 3 is 5x + (−3), so its terms are 5x and −3.

Each term also has a degree, the number of variable factors in it. 5x has degree 1, 5x² has degree 2 and the constant 6 has degree 0. The degree of an expression is the highest degree among its terms, so 3x² − 5x + 1 has degree 2. An expression whose highest degree is 1, such as 2x + 7, is called linear. You will meet linear expressions in every chapter that follows, because linear equations and straight-line graphs are built from them.

Like terms carry the same letters to the same powers

Terms are like terms if they have the same variables with the same exponents, and all constant terms are like terms. Among 5x, 7, n², 4, 3x and 9n², the like terms pair up as 5x with 3x, n² with 9n², and 7 with 4. The coefficients do not matter for this test, only the letters and their powers. 2x and 2x² are not like terms, and neither are x and xy. The tiles below show why. An x² tile is a big square, an x tile is a long strip and a 1 tile is a small square, and only tiles of the same shape can be counted together.

Algebra tiles. Blue squares are x², green strips are x and orange squares are 1. Two groups, 2x² + 3x + 4 and x² + 2x + 1, are sorted by shape: 3 squares, 5 strips and 5 small squares, so the sum is 3x² + 5x + 5.

Combining like terms

To simplify, add the coefficients of like terms and keep the letters as they are. In 3x + 7 + 4x + 5, move the like terms next to each other, which addition allows because order does not change a sum: 3x + 4x + 7 + 5 = 7x + 12. Negative coefficients work the same way with subtraction. OpenStax simplifies 7x² + 8x − x² − 4x to 6x² + 4x, and stops there, because 6x² and 4x are not like terms. Two expressions can be combined the same way: (5x + 3) + (2x − 4) is 7x − 1. A minus sign in front of a bracket flips the sign of every term inside it, so (5x + 3) − (2x − 4) is 5x + 3 − 2x + 4 = 3x + 7. The next unit explains why with the distributive property.

  1. Which term is a like term of 5x?
    • −2x
    • 5x²
    • 5
    • 5y
  2. What is the coefficient of a in 9b + 15x² + a + 6?
  3. What is the coefficient of x in 4 − x?
  4. What is the degree of 3x² − 5x + 1?
  5. Simplify 3x + 6x.
  6. Simplify 3x + 7 + 4x + 5.
  7. Simplify 7x² + 8x − x² − 4x.
  8. Simplify 2a + 3b − a + 4b.
  9. Simplify (5x + 3) − (2x − 4).
  10. After combining like terms in 4x² + 2x + 5x² + 6x + 40x + 8xy, how many terms are left?
  11. 3x + 5x and 8x² both equal 8 when x = 1. Are they equivalent?
    • No: at x = 2 they give 16 and 32
    • Yes: they agree at x = 1
    • Yes: both have coefficient 8
    • Only when x is positive
Common mistakes
  • “3x + 5 = 8x.” 3x and 5 are not like terms, so they cannot be added. At x = 2 the left side is 11 and 8x is 16. The expression 3x + 5 is already as simple as it gets.
  • “2x + 3x = 5x², because x and x make x².” Adding counts the x’s; it does not multiply them. 2x + 3x = 5x. Only x · x is x².
  • “5x − (2x − 4) = 3x − 4.” The minus in front of the bracket applies to both terms inside: −(2x − 4) = −2x + 4. The answer is 3x + 4.
Teacher's noteIn the 2022 Korean national curriculum, adding and subtracting linear expressions is a middle school standard that asks learners to understand the principle as well as do the calculation, and a later standard extends it to polynomials. The term list for this area includes term, constant term, coefficient, degree, linear expression and like terms. Tiles make the principle visible, and a spreadsheet like the one above does the same with numbers. Learners who see 8x² agree with 3x + 5x at x = 1 and fail at x = 2 remember the lesson that one example never proves two expressions equal. Ask them to pick their own test values, and steer them away from 0 and 1.
Key points
  • Equivalent expressions give the same value for every x; one counterexample shows they differ.
  • Terms are joined by + (read − as adding a negative). The number in front is the coefficient.
  • Like terms have the same letters to the same powers; constants are like terms.
  • Combine like terms by adding coefficients: 3x + 7 + 4x + 5 = 7x + 12.
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