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➕ Algebra 1 · Exponents and Polynomials

Adding and Subtracting Polynomials: notes and common mistakes

Study notes

Key ideas

  • A polynomial is a monomial or a sum of monomials, with whole-number exponents. Monomial, binomial and trinomial have 1, 2 and 3 terms.
  • The degree of a polynomial is the highest degree among its terms (with one variable, that is the highest exponent; for a term with several variables, add their exponents). Standard form writes the terms from the highest degree to the lowest.
  • Adding polynomials means combining like terms: add the coefficients and keep the variable part.
  • A minus sign in front of a polynomial flips the sign of every term inside it. Subtracting is the same as adding the opposite.
  • The sum or difference of two polynomials is again a polynomial. Its degree can be lower than the degrees you started with.

Steps

  1. Write each polynomial in standard form (use a 0 placeholder for any missing power in vertical form).
  2. For subtraction, flip the sign of every term in the second polynomial and change the operation to addition.
  3. Group like terms and add their coefficients.
  4. Write the answer in standard form.
  5. Check with x = 3 in the original and in your answer.

Formulas

  • Adding: (ax2 + bx + c) + (dx2 + ex + f) = (a + d)x2 + (b + e)x + (c + f)
  • Subtracting: (ax2 + bx + c) − (dx2 + ex + f) = (a − d)x2 + (b − e)x + (c − f)
  • Minus sign in front: −(ax2 + bx + c) = −ax2 − bx − c
  • Subtracting is adding the opposite: A − B = A + (−B)
  • Degree of a sum or difference: At most the larger of the two degrees (it can be smaller if leading terms cancel)

Watch out

  • The minus sign flips every term of the second polynomial: −(3x − 8) = −3x + 8, not −3x − 8.
  • Adding like terms changes the coefficient, not the exponent: 5x3 + 2x3 = 7x3, not 7x6.
  • Unlike terms do not combine: 2x2 + 6x stays as it is.
  • The degree is the highest degree among the terms (with one variable, the highest exponent), not the degree of the first term written. Write the polynomial in standard form to see it.
  • Test with x = 3. The values 0, 1 and 2 can hide mistakes, and one matching value is a good sign, not a proof.

Why it works

  • Why can you combine like terms? 4x2 + 2x2 = (4 + 2)x2 is the distributive property read backward: 4 of something plus 2 of the same something is 6 of it.
  • Why does the minus sign flip every term? −(P) means (−1)(P), and the distributive property multiplies −1 by every term inside the parentheses.
  • Why is the result a polynomial? Adding or subtracting only changes coefficients, and the exponents stay whole numbers. That is what “closed under addition and subtraction” means.

Common mistakes

✗ Mistake(6x − 5) − (3x − 8) = 3x − 13 (only the first term of the second group is subtracted).
✓ InsteadThe minus sign applies to every term of the second group: −(3x − 8) = −3x + 8. So the answer is 6x − 5 − 3x + 8 = 3x + 3.
✗ Mistake5x3 + 2x3 = 7x6 (adding the exponents when adding).
✓ InsteadAdding like terms adds the coefficients and keeps the exponent: 5x3 + 2x3 = 7x3. Only multiplying adds exponents.
✗ Mistake2x2 + 6x = 8x3 (combining unlike terms).
✓ Insteadx2 and x are different variable parts, so they are not like terms. 2x2 + 6x is already as simple as it gets.
✗ MistakeThe degree of 3x + 8x5 is 1, because 3x comes first.
✓ InsteadWith one variable, the degree is the highest exponent, which is 5. In standard form, 8x5 + 3x, the highest power comes first.
✗ MistakeChecking with x = 0 or x = 1 is enough to prove an answer is right.
✓ InsteadAt x = 0 only the constants are tested, and at x = 1 every power of x equals 1, so both can hide mistakes. Use a value like 3. One match is a good sign, not a proof.
✗ MistakeAdding two polynomials of degree n always gives a polynomial of degree n.
✓ InsteadThe degree can drop when leading terms cancel: (x2 + 3x) + (−x2 + 4) = 3x + 4 has degree 1.

Teach it back

Explain to a friend how to subtract (8x2 − 3x + 2) − (3x2 + 5x − 6), why every sign in the second polynomial flips, and how to check your answer with x = 3.

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

  • I wrote −(3x2 + 5x − 6) = −3x2 − 5x + 6, flipping all three signs, and said the minus sign means multiplying by −1.
  • I combined like terms: 8x2 − 3x2 = 5x2, −3x − 5x = −8x and 2 + 6 = 8, to get 5x2 − 8x + 8.
  • I checked with x = 3: the original is 65 − 36 = 29 and the answer is 45 − 24 + 8 = 29.

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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