Addition & Subtraction of Polynomials: Like Terms Guide

A polynomial is an expression made of terms with whole-number exponents, like 3x² + 5x − 2. Adding and subtracting polynomials is just combining like terms on a bigger scale: line up the x² terms together, the x terms together, and the constants together, then merge each group.

This operation is the arithmetic of algebra — every equation you solve, every identity you expand, and every polynomial you divide starts with adding or subtracting them correctly. Engineers adding load formulas and economists combining cost models both do exactly this.

Key Concepts

• Polynomial: a sum of terms where each variable has a whole-number exponent (x², x³ are fine; 1/x is not).
• Degree: the highest exponent in the polynomial — 3x² + 5x − 2 has degree 2 (quadratic).
• Standard form: terms written from highest to lowest degree.
• Adding polynomials: remove brackets and combine like terms.
• Subtracting polynomials: flip the sign of every term in the subtracted polynomial, then combine like terms.

How It Works

Add (3x² + 5x − 2) + (x² − 3x + 7):

1. Drop the brackets (a plus sign changes nothing): 3x² + 5x − 2 + x² − 3x + 7.
2. Group like terms: x²-terms (3x², x²), x-terms (5x, −3x), constants (−2, 7).
3. Combine: 3x² + x² = 4x²; 5x − 3x = 2x; −2 + 7 = 5.
4. Answer: 4x² + 2x + 5.

For subtraction, (3x² + 5x − 2) − (x² − 3x + 7): flip every sign in the second bracket to get 3x² + 5x − 2 − x² + 3x − 7, then combine: 2x² + 8x − 9.

Worked Example

Simplify: (4y³ − 2y² + y) + (y³ + 5y² − 3y + 1).

1. Remove brackets: 4y³ − 2y² + y + y³ + 5y² − 3y + 1.
2. Group: y³-terms (4y³, y³), y²-terms (−2y², 5y²), y-terms (y, −3y), constant (1).
3. Combine: 4y³ + y³ = 5y³; −2y² + 5y² = 3y²; y − 3y = −2y; constant stays 1.

The sum is 5y³ + 3y² − 2y + 1. Verify with y = 1: first polynomial gives 4 − 2 + 1 = 3; second gives 1 + 5 − 3 + 1 = 4; total 7. The simplified form at y = 1: 5 + 3 − 2 + 1 = 7. Matches.

Common Misconceptions

• "Subtracting flips only the first sign." The most expensive mistake in algebra. In −(x² − 3x + 7), all three terms change sign: −x² + 3x − 7. Treat the minus as multiplying the whole bracket by −1.
• "x² + x = x³." Exponents add only when multiplying, never when adding. x² + x stays as x² + x — they are unlike terms.
• "A term with no partner can be skipped." In subtraction, a term with no partner still counts: in the worked example, the constant 1 had no partner in the first polynomial but survives into the answer.

Real-World Applications

• Physicists add polynomial models of motion and forces.
• Economists combine revenue and cost polynomials to get profit functions.
• Computer graphics uses polynomial arithmetic for curves and surfaces.
• Signal processing builds filters by adding polynomial expressions.

Adding polynomials means dropping brackets and merging like terms; subtracting means flipping every sign first, then merging. Master the sign flip, and polynomial arithmetic becomes automatic.