💡 The Two Key Skills
To simplify polynomial expressions, you need two things: (1) correctly distribute a minus sign across parentheses — flipping every sign inside — and (2) combine only like terms — terms that have the exact same variable(s) and exponent(s). Mix either skill up and you'll get a wrong answer!
Like terms have the same variable part. You can add or subtract like terms by combining their coefficients.
Unlike terms have different variable parts. You CANNOT combine them — they stay separate!
a and ab — different (one has b, the other doesn't)
a and b — different variables
ab and b — different (ab has two variables)
a and a² — different exponents
| Group | From 1st expression | From 2nd expression |
|---|---|---|
| a-terms | 3a | a |
| ab-terms | 4ab | 2ab |
| b-terms | −7b | −4b |
When you subtract a polynomial in parentheses, distribute the −1 to every term inside:
−(A + B − C) = −A − B + C
Every sign flips: + becomes −, and − becomes +!
−(+a) = −a −(+2ab) = −2ab
−(−4b) = +4b −(−7) = +7
Students often flip the first term's sign but forget to flip the last term when it's negative.
−(a + 2ab − 4b) → the −4b becomes +4b. Many write −4b instead — this gives −11b instead of −3b!
= 3a + 4ab − 7b − a − 2ab + 4b
= (3a − a) + (4ab − 2ab) + (−7b + 4b)
= 2a + 2ab − 3b
= 2a + 2ab + (−3b)
| Choice | What Went Wrong | Correct? |
|---|---|---|
| 2a + 2ab − 11b | Forgot to flip −4b → left it as −4b: −7b−4b = −11b ❌ | ❌ Trap! |
| 2a + 6ab − 11b | Added ab-terms instead of subtracting AND −11b error ❌ | ❌ Wrong |
| 2a + 2ab − 3b | All signs distributed correctly ✅ | ✅ CORRECT |
| 2a + 6ab − 3b | Added 4ab+2ab=6ab instead of subtracting ❌ | ❌ Wrong |
= 2x + 5y + 3x − 2y
= 3x² + 4x − 6
= 3m + 2mn − 3n
Problem: −(a + 2ab − 4b)
Wrong: −a − 2ab − 4b → gives −7b − 4b = −11b
Correct: −a − 2ab + 4b → gives −7b + 4b = −3b ✅
The answer 2a + 2ab − 11b comes from this exact mistake!
After distributing: 4ab − 2ab
Wrong: 4ab + 2ab = 6ab
Correct: 4ab − 2ab = 2ab ✅
The answer 2a + 6ab − 3b comes from treating −2ab as +2ab!
Never combine: 2a + 2ab (a and ab are NOT like terms!)
Never combine: 4ab − 3b (ab and b are NOT like terms!)
Your final answer must keep all three groups separate: a-terms, ab-terms, b-terms.
Underline all a-terms in blue, circle ab-terms in orange, and box b-terms in red. Then combine only what matches color. This eliminates Trap 3!
Change every sign inside the subtracted parentheses. + → −, and − → +. Write out the full expanded expression.
Underline all a-terms, circle ab-terms, box b-terms. Visual grouping prevents mixing unlike terms.
3a − a = (3−1)a = 2a. Treat the variable part as a label and just operate on the numbers in front.
Typically: highest-degree terms first, then lower. For multi-variable: match the order of the original expression.
Original: (3+4−7)−(1+2−4) = 0−(−1) = 1. Answer: 2+2−3 = 1 ✅ Both give 1!