🧱 See It
A polynomial is several terms added or subtracted together, like 3x² + 2x − 5. "Like terms" have the exact same variable part — same letter AND same exponent. Only like terms can ever be combined; 3x² and 2x can never merge into one term, no matter what.
✏️ Draw It
Stack the two polynomials so matching like-terms line up in columns — an x² column, an x column, and a constant column, exactly like lining up decimal place values:
| x² column | x column | constant column |
|---|---|---|
3x² | 2x | −5 |
x² | −4x | 1 |
Only add down each column — never across columns.
🔢 Write It
(3x² + 2x − 5) + (x² − 4x + 1)
Combine each column separately:
x²column:3x² + x² = 4x²xcolumn:2x − 4x = −2x- constant column:
−5 + 1 = −4
Final: 4x² − 2x − 4
Subtracting a polynomial means distributing a negative sign across EVERY one of its terms first — reusing the "subtracting a negative flips the sign" idea from integers.
💡 Worked Examples
Example 1 — Add: (2x² + 3x + 4) + (x² − x − 6)
x²column:2x² + x² = 3x²xcolumn:3x − x = 2x- constant column:
4 − 6 = −2 - Answer:
3x² + 2x − 2
Example 2 — Subtract: (5x² + 3x) − (2x² − x + 4)
- Distribute the negative across every term first:
5x² + 3x − 2x² + x − 4 x²column:5x² − 2x² = 3x²xcolumn:3x + x = 4x- constant column:
−4(nothing else to combine with) - Answer:
3x² + 4x − 4
🧠 Quick Reference
Only combine LIKE terms — same variable, same exponent. Different exponents (like x² and x) never combine, no matter how tempting it looks. Subtracting a polynomial? Distribute the negative sign across every term in it FIRST.