Adding two polynomials is genuinely just combining like terms — the same operation used throughout Polynomials — but longer polynomials make it easy to miss a term or combine two terms that only look alike. This lesson isolates the addition case specifically and drills the one habit that prevents both mistakes: identify every like term first, then combine.
What Are Like Terms?
Like terms share the exact same variable part, including every exponent. Only the coefficient is allowed to differ.
$$ 3x^{2} \text{ and } -7x^{2} \;\Rightarrow\; \text{like terms (both } x^{2}\text{)} \qquad 3x^{2} \text{ and } 3x^{3} \;\Rightarrow\; \text{not like terms (different exponents)} $$
Combining like terms adds their coefficients and leaves the variable part completely unchanged:
$$ 3x^{2} + (-7x^{2}) = (3+(-7))x^{2} = -4x^{2} $$
Adding Polynomials Horizontally
To add two polynomials written left to right, drop the parentheses (addition never changes any sign inside them), then group and combine like terms.
$$ (3x^{2}-5x+7) + (2x^{2}+4x-3) $$
$$ = 3x^{2}-5x+7+2x^{2}+4x-3 $$
$$ = (3x^{2}+2x^{2}) + (-5x+4x) + (7-3) = 5x^{2} - x + 4 $$
Grouping by matching exponent (all the \(x^{2}\) terms, then all the \(x\) terms, then all the constants) before combining is what keeps a longer sum from mixing up terms that only superficially look similar.
Adding Polynomials Vertically
For polynomials with several terms, stacking them so matching exponents line up in the same column makes every like term visually obvious, and combining becomes column-by-column addition.
$$ \begin{array}{r} 3x^{3} - 2x^{2} + 5x - 1 \\ +\; (x^{3} + 4x^{2} - 3x + 6) \\ \hline 4x^{3} + 2x^{2} + 2x + 5 \end{array} $$
Each column adds its own coefficients independently: \(3+1=4\) for the \(x^{3}\) column, \(-2+4=2\) for the \(x^{2}\) column, and so on. If one polynomial is missing a term that the other has, leave that column’s spot blank (effectively a coefficient of \(0\)) rather than shifting the remaining terms out of alignment.
Adding Polynomials in More Than One Variable
With two or more variables, the “like terms” check needs every variable’s exponent to match, not just one of them.
$$ (3x^{2}y - 4xy^{2} + 2) + (5x^{2}y + xy^{2} - 6) $$
$$ = (3x^{2}y+5x^{2}y) + (-4xy^{2}+xy^{2}) + (2-6) = 8x^{2}y - 3xy^{2} - 4 $$
\(x^{2}y\) and \(xy^{2}\) are not like terms, even though both involve \(x\) and \(y\) — the exponents are attached to different variables in each, so they stay as two separate terms in the sum rather than combining with each other.
Worked Example A: Adding Two Trinomials Horizontally
Add \((4x^{2}+3x-2) + (x^{2}-7x+5)\).
Step 1 — remove the parentheses:
$$ 4x^{2}+3x-2+x^{2}-7x+5 $$
Step 2 — group like terms:
$$ (4x^{2}+x^{2}) + (3x-7x) + (-2+5) $$
Step 3 — combine each group:
$$ 5x^{2} - 4x + 3 $$
Answer: \(5x^{2}-4x+3\)
Worked Example B: Adding Vertically with a Missing Term
Add \((2x^{3}-5x+9) + (x^{3}+3x^{2}-4)\) using vertical alignment.
Step 1 — stack the polynomials, leaving a gap where a term is missing:
$$ \begin{array}{r} 2x^{3} \phantom{+ 3x^{2}} - 5x + 9 \\ +\; (x^{3} + 3x^{2} \phantom{- 5x} - 4) \\ \hline 3x^{3} + 3x^{2} - 5x + 5 \end{array} $$
Step 2 — read off the sum column by column: \(2+1=3\) for \(x^{3}\); only the second polynomial has an \(x^{2}\) term, so it passes through as \(3x^{2}\); only the first has an \(x\) term, so it passes through as \(-5x\); the constants combine as \(9+(-4)=5\).
Answer: \(3x^{3}+3x^{2}-5x+5\)
Worked Example C: Adding Three Polynomials at Once
Add \((x^{2}+2x) + (3x^{2}-x+4) + (-2x^{2}+5)\).
Step 1 — remove parentheses and group like terms:
$$ (x^{2}+3x^{2}-2x^{2}) + (2x-x) + (4+5) $$
Step 2 — combine each group:
$$ 2x^{2} + x + 9 $$
Answer: \(2x^{2}+x+9\). Grouping first works exactly the same way whether two polynomials are being added or several — every like term still gets found before anything is combined.
Worked Example D: Adding Polynomials in Two Variables
Add \((7x^{2}y-3xy+2y^{2}) + (-2x^{2}y+3xy-5y^{2})\).
Step 1 — group like terms (matching both variables and both exponents):
$$ (7x^{2}y-2x^{2}y) + (-3xy+3xy) + (2y^{2}-5y^{2}) $$
Step 2 — combine each group:
$$ 5x^{2}y + 0 - 3y^{2} = 5x^{2}y - 3y^{2} $$
Answer: \(5x^{2}y-3y^{2}\). The \(xy\) terms cancelled to \(0\) and simply don’t appear in the final answer.
Common Mistakes to Avoid
- Combining terms that only look similar. \(3x^{2}\) and \(5x\) are not like terms — the exponents differ — and can never be combined into a single term.
- Adding exponents instead of coefficients. \(3x^{2}+5x^{2}=8x^{2}\), not \(8x^{4}\); only the coefficients change when like terms combine.
- Losing a term when one polynomial is missing it. A missing term acts like a coefficient of \(0\), not like the alignment should shift to skip that column.
- Mixing up which variable an exponent belongs to. \(x^{2}y\) and \(xy^{2}\) look similar but are not like terms, since the \(2\) applies to a different variable in each.
- Forgetting a negative sign belongs to the term that follows it. In \(3x^{2}-5x+7\), the \(-5x\) term carries a coefficient of \(-5\), not \(5\) with a separate subtraction floating nearby.
- Skipping the grouping step on a longer sum. Combining terms in the order they happen to appear, rather than grouping by matching exponent first, is the most common source of an accidentally-skipped or double-counted term.
Where This Shows Up Later
- Subtracting Polynomials. Subtraction is addition of the opposite polynomial, so every technique here — grouping, vertical alignment, handling missing terms — carries over directly.
- Multiplying Polynomials. Multiplying two polynomials produces several partial products that then have to be added together using exactly this like-terms process.
- Factoring Polynomials. Recognizing that a factored expression expands back to the original polynomial requires being able to add the resulting terms correctly as a check.
- Functions. Adding two polynomial functions, written \((f+g)(x)\), is defined as adding their outputs, which is this same term-by-term combination applied to two function rules instead of two standalone polynomials.
Practice Problems
Work each problem yourself before opening the answer.
Problem 1. Add \((5x^{2}+3x-1) + (2x^{2}-4x+6)\).
Show answer
$$ (5x^{2}+2x^{2}) + (3x-4x) + (-1+6) = 7x^{2} - x + 5 $$
Answer: \(7x^{2}-x+5\)
Problem 2. Add \((4x^{3}-2x+7) + (x^{3}+5x^{2}-3)\).
Show answer
$$ (4x^{3}+x^{3}) + 5x^{2} + (-2x) + (7-3) = 5x^{3} + 5x^{2} - 2x + 4 $$
Answer: \(5x^{3}+5x^{2}-2x+4\)
Problem 3. Add \((-3x^{2}+6x-2) + (3x^{2}-6x+9)\).
Show answer
$$ (-3x^{2}+3x^{2}) + (6x-6x) + (-2+9) = 0 + 0 + 7 = 7 $$
Answer: \(7\)
Problem 4. Add \((2x^{2}+3xy-y^{2}) + (x^{2}-xy+4y^{2})\).
Show answer
$$ (2x^{2}+x^{2}) + (3xy-xy) + (-y^{2}+4y^{2}) = 3x^{2} + 2xy + 3y^{2} $$
Answer: \(3x^{2}+2xy+3y^{2}\)
Problem 5. Add \((6x^{4}-3x^{2}+8) + (2x^{4}+5x^{3}-x)\).
Show answer
$$ (6x^{4}+2x^{4}) + 5x^{3} + (-3x^{2}) + (-x) + 8 = 8x^{4} + 5x^{3} - 3x^{2} - x + 8 $$
Answer: \(8x^{4}+5x^{3}-3x^{2}-x+8\)
Problem 6. Add \((x^{2}-5x+3) + (-x^{2}+5x-3)\).
Show answer
$$ (x^{2}-x^{2}) + (-5x+5x) + (3-3) = 0 $$
Answer: \(0\)
Problem 7. Add \((3x^{3}+2x^{2}-7) + (4x^{2}-x+5) + (x^{3}-3)\).
Show answer
$$ (3x^{3}+x^{3}) + (2x^{2}+4x^{2}) + (-x) + (-7+5-3) = 4x^{3} + 6x^{2} - x - 5 $$
Answer: \(4x^{3}+6x^{2}-x-5\)
Problem 8. Add \((5x^{2}y-2xy^{2}) + (-3x^{2}y+7xy^{2})\).
Show answer
$$ (5x^{2}y-3x^{2}y) + (-2xy^{2}+7xy^{2}) = 2x^{2}y + 5xy^{2} $$
Answer: \(2x^{2}y+5xy^{2}\)
Problem 9. Add \((7x-4) + (2x^{2}+3x-1) + (-x^{2}+2)\).
Show answer
$$ (2x^{2}-x^{2}) + (7x+3x) + (-4-1+2) = x^{2} + 10x - 3 $$
Answer: \(x^{2}+10x-3\)
Problem 10. Add \((4x^{2}-3x+9) + (0)\) — that is, add the zero polynomial.
Show answer
Adding \(0\) to any polynomial leaves it completely unchanged, since \(0\) contributes nothing to any column:
$$ 4x^{2}-3x+9 $$
Answer: \(4x^{2}-3x+9\)
Quick Reference
| Situation | Move |
|---|---|
| Two terms have the same variable part | Add their coefficients; the variable part stays the same |
| Two terms have different exponents (or different variables) | They are not like terms — leave both as separate terms |
| Adding horizontally | Drop parentheses, then group all like terms together before combining |
| Adding vertically | Align matching exponents in the same column; leave a blank for any missing term |
| A term is missing from one polynomial | Treat it as a coefficient of \(0\) in that column |
| More than one variable | Every variable’s exponent must match for two terms to be alike |
Once combining like terms feels automatic in both directions, Subtracting Polynomials applies the identical grouping habit with one extra sign-distribution step, and Multiplying Polynomials uses this addition process to combine the partial products of a multiplication. Revisit Polynomials for the full overview these three lessons split apart. Build speed with the Adding Polynomials Generator, or browse the rest of the Algebra lessons as new ones publish.