Adding Polynomials: Rules, Methods, Worked Examples

TL;DR

Adding polynomials means combining like terms: terms with the same variable raised to the same power. You can add them horizontally, grouping like terms in a line, or vertically, stacking them in columns by power. This article covers the like-terms rule, both methods step by step, what happens with unlike terms, the sign and exponent traps that cost marks, and six worked examples from two binomials to multivariable polynomials.

What Does Adding Polynomials Mean?

Adding polynomials is the process of combining two or more polynomials into a single one by summing their like terms. A polynomial is a sum of terms; each term has a coefficient and a variable part. When you add polynomials, the variable parts that match get their coefficients added, and the rest stay as they are.

The result is always another polynomial, and its degree is at most the highest degree among the polynomials you added. Adding never raises the degree.

Variable Glossary:

What Are Like Terms?

Like terms are terms whose variables and exponents are identical; only the coefficient may differ. The terms 4x² and −7x² are like terms because both carry x². The terms 4x² and 4x are unlike terms, even though both use x, because the powers differ.

This is the rule the whole operation depends on:

• Like terms combine. Add the coefficients, keep the variable part. 4x² + (-7x²) = -3x².
• Unlike terms never combine. They are simply carried into the answer unchanged. 4x² + 4x stays as 4x² + 4x.

Spotting like terms in a multivariable expression takes a second look. The terms 3xy and 5xy are like; 3xy and 3x²y are not, because the exponent on x differs.

How Do You Add Polynomials Horizontally?

The horizontal method writes the polynomials in a line with a plus sign between them, groups the like terms, and adds. It suits short polynomials you can scan in one read.

Add 5x² + 3x - 2 and 3x² - x + 4:

Write them in a line: (5x² + 3x - 2) + (3x² - x + 4).

Group like terms: (5x² + 3x²) + (3x - x) + (-2 + 4).

Add each group:

5x² + 3x² = 8x²

3x - x = 2x

-2 + 4 = 2

Result: 8x² + 2x + 2.

How Do You Add Polynomials Vertically?

The vertical method stacks the polynomials in columns, lining up like terms under each other, then adds down each column. It scales better for long polynomials and is where missing-power gaps matter.

Add 2x² + 3x + 2 and 3x² - 5x - 1, aligned by power:

[\begin{array}{r} 2x^2 + 3x + 2 \ 3x^2 - 5x - 1 \end{array}]

Add down each column:

2x² + 3x² = 5x²

3x - 5x = -2x

2 - 1 = 1

Result: 5x² - 2x + 1.

Properties of Polynomial Addition

Polynomial addition obeys the same three structural properties as ordinary number addition:

• Closure. The sum of two polynomials is always another polynomial.
• Commutative. The order does not matter: P + Q = Q + P.
• Associative. The grouping does not matter: (P + Q) + R = P + (Q + R).

There is also an additive identity, the zero polynomial: adding 0 to any polynomial leaves it unchanged.

Examples of Adding Polynomials

Example 1

Add (3x + 5) and (2x + 1).
Result: 5x + 6.

Example 2

Add (4x² + 3x) and (2x² + 5x).
Result: 6x² + 8x.

Example 3

Add (5x² + 3x - 2) and (3x² - x + 4).
Result: 8x² + 2x + 2.

Example 4

Add (2x³ + 4x - 1) and (x³ - 2x² + 5).
Result: 3x³ - 2x² + 4x + 4.

Example 5

Add the multivariable polynomials (2x² + 5xy + 4y²) and (3x² - 2xy + y²).
Result: 5x² + 3xy + 5y².

Example 6

Add three polynomials: (x² + 2x + 1), (2x² - x + 3), and (-x² + 4x - 5).
Result: 2x² + 5x - 1.

Key Takeaways

• Adding polynomials means combining like terms by adding their coefficients while keeping the variable part unchanged.
• Like terms share the same variable and the same exponent; unlike terms never combine.
• The horizontal method groups like terms in a line; the vertical method stacks them in columns by power.
• Add the coefficients, never the exponents, and insert zero placeholders for missing powers in the vertical method.
• The result of adding polynomials is always another polynomial whose degree never exceeds the inputs.