1
Like terms are terms with the identical variable part
3x² and −7x² are like terms. 3x² and 3x³ are not, and no amount of rearranging will make them so. Same variable, same exponent — both conditions, every time.
Lesson · Polynomials and Factoring · Polynomial arithmetic
Adding and Subtracting Polynomials
Addition collects like terms, and the licence to do it is the distributive law read backwards. Subtraction is the same operation after the minus sign has been distributed over every term — which is where the errors live.
axⁿ + bxⁿ = (a + b)xⁿ like terms combine
x² + x³ stays as it is NOT like terms
A − (B + C) = A − B − C the minus reaches every term
One legal move, and one trap
Core ideas
What this lesson establishes
There is only one new idea here, and it is not "add the numbers". It is the question of when you are allowed to.
2
Combining like terms IS the distributive law
3x² + 5x² = (3 + 5)x² = 8x² because ab + ac = a(b + c) with a = x². You are not inventing a rule for polynomials; you are using the one rule arithmetic already gave you.
3
Subtraction distributes over every term, not just the first
(A) − (B + C) is A − B − C. Missing the second sign flip is the most common single error in this course, and it produces an answer that looks entirely reasonable.
The material
Collecting what can be collected
Why like terms combine, why unlike terms cannot, and what subtraction really is.
Definition · Like terms
Two terms are like terms when their variable parts are identical — the same variables raised to the same exponents. 5x²y and −2x²y are like terms; 5x²y and 5xy² are not.
The coefficients play no part in deciding. 100x³ and x³ are like terms; x³ and x² are not, however close the exponents look.
Theorem · Why like terms combine
For any number a, b and any whole number n, axⁿ + bxⁿ = (a + b)xⁿ.
This is the distributive law ac + bc = (a + b)c with c = xⁿ. It is not a special rule about polynomials, and that is exactly why it is safe.
Turn the theorem around and you can see why x² + x³ cannot be collected: there is no common c to pull out other than x², and pulling that out gives x²(1 + x) — a product, not a shorter sum. It is a legitimate move, and it is factoring, which starts in “Factoring Out the Greatest Common Factor”. It just is not addition.
Subtraction
Subtracting a polynomial means adding its negative, and the negative of a polynomial has every sign flipped:
−(x² − 5x − 2) = −x² + 5x + 2
not −x² − 5x − 2
The left-hand form is the distributive law with −1: −1 · (x² − 5x − 2) = (−1)x² + (−1)(−5x) + (−1)(−2). Nothing is being remembered here; every sign is computed.
The safest habit is to write the negated polynomial out as its own line before adding anything. It costs one line and removes the error class entirely.
What happens to the degree
The degree of a sum is at most the larger of the two degrees. Usually it equals it — but if the leading terms cancel, the degree drops:
(5x⁴ − 2x² + x) + (−5x⁴ + 2x² + 3) = x + 3
degree 4 + degree 4 → degree 1
This is why “Polynomials, Degree and Standard Form” insisted that degree is read after combining. Addition can lower a degree; it can never raise one.
Example · Adding in one line
(3x³ − 5x + 2) + (x³ + 4x² − 5). The x³ terms give 4x³; there is only one x² term, 4x²; only one x term, −5x; the constants give 2 − 5 = −3. Result: 4x³ + 4x² − 5x − 3.
Interactive
Adding and subtracting
Like terms, lined up by power
Method
Adding or subtracting, without sign errors
For subtraction, step 1 is the whole battle.
1
If it is a subtraction, negate the second polynomial first
Write −(…) out in full with every sign flipped, on its own line, before you collect anything. Then the problem is an addition and stays one.
2
Put both polynomials in standard form
Descending degree, both of them. Like terms then line up in columns and you can see at a glance which powers are present.
3
Collect one power at a time, highest first
Take all the x³ terms, then all the x² terms, and so on. Working by power rather than left to right is what stops a term being used twice or skipped.
4
Read the degree off the answer, not off the inputs
If the leading terms cancelled, the answer has a lower degree than either input. That is a legitimate result, not a mistake.
Collect and subtract
Score 0 / 0Choose the answer you can justify.
One subtraction, in full
(4x² − 3x + 6) − (x² − 5x − 2), with the negation written out.
step 1 negate the second polynomial
−(x² − 5x − 2) = −x² + 5x + 2
step 2 now it is an addition
(4x² − 3x + 6) + (−x² + 5x + 2)
step 3 collect by power
x² : 4 + (−1) = 3 → 3x²
x : −3 + 5 = 2 → 2x
1 : 6 + 2 = 8 → 8
answer 3x² + 2x + 8
The two signs that flipped were on −5x and −2, and both became positive. Skipping step 1 and writing 4x² − 3x + 6 − x² − 5x − 2 gives 3x² − 8x + 4: a tidy-looking answer, and wrong in two places.
Check any result by substituting a number into the original and into the answer. At x = 1 the original is 7 − (−6) = 13, and the answer is 3 + 2 + 8 = 13. One substitution catches nearly every sign error.
For a faded subtraction, begin (3x³ − 2x + 4) − (−x³ + 5x − 6) with the supplied negation +x³ − 5x + 6. Collect the three powers and check at x = 1. You should obtain 4x³ − 7x + 10; the original and answer both give 7 at x = 1. A middle term of +3x means the 5x was not negated.
Common mistakes
1
Distributing the minus sign to the first term onlyA − (B + C) is A − B − C. This single error accounts for more wrong answers than every other mistake in the course combined, which is why the negation deserves its own line.
2
Adding the exponents when adding like terms3x² + 5x² = 8x², not 8x⁴. Exponents add when terms are multiplied — that is “Multiplying Polynomials”, and mixing the two rules is easy once both are in your hands.
3
Forcing unlike terms togetherx² + x³ is finished; it is not x⁵ and not 2x⁵. If the variable parts differ at all, the sum is already as short as it gets.
Check your understanding
✓
Finish when a subtraction is as safe for you as an addition.
You should be able to take a difference of two quartics and get every sign right first time, because you write the negation on its own line rather than doing it in your head. Substituting x = 1 into both the question and your answer should be automatic.
Note: The x = 1 check is worth making a habit now. It costs seconds, it catches sign errors and dropped terms, and it is the same idea — evaluating a polynomial at a number — that becomes the remainder theorem in “Synthetic Division and the Remainder Theorem”.