Adding and subtracting polynomials

Adding polynomials

To add polynomials, remove the brackets and combine like terms.

Example: (3x² + 2x − 5) + (x² − 6x + 8)

• x² terms: 3x² + x² = 4x²
• x terms: 2x − 6x = −4x
• Constants: −5 + 8 = 3

Answer: 4x² − 4x + 3

Only like terms combine

Add coefficients; keep the variable part unchanged. x² + x² = 2x², not x⁴.

Subtracting polynomials

Subtracting means adding the opposite: change the sign of every term in the second polynomial.

Example: (5x² − 3x + 4) − (2x² + x − 7)

1. Distribute the negative: 5x² − 3x + 4 − 2x² − x + 7
2. Combine: 3x² − 4x + 11

The most common mistake is changing only the first sign.

Vertical method

Line up like terms in columns:

   5x² − 3x + 4
 − (2x² +  x − 7)
 ----------------
   3x² − 4x + 11

Leave a gap when a power is missing.

Writing the answer

Write the result in standard form: highest degree first.

• 7 + 2x³ − x → 2x³ − x + 7

Polynomials with several variables

Example: (4ab − 2a + b) + (−ab + 5a − 3b) = 3ab + 3a − 2b

Perimeter application

A triangle has sides (2x + 3), (x − 1) and (3x + 5). Perimeter:

(2x + 3) + (x − 1) + (3x + 5) = 6x + 7

Worked examples

Example 1: (6y² + y) + (2y² − 4y + 1) → 8y² − 3y + 1.

Example 2: (x² + 5) − (3x² − 2x + 5) → x² + 5 − 3x² + 2x − 5 → −2x² + 2x.

Example 3: Subtract 4m − 7 from 9m + 2 → (9m + 2) − (4m − 7) → 5m + 9.

Common mistakes

• Not distributing the minus sign to every term.
• Combining unlike terms.
• Adding exponents when adding terms.
• Subtracting in the wrong order for "subtract A from B" (it’s B − A).

Try it yourself

1. (2x² + 3x) + (5x² − x − 4)
2. (7a − 3) − (2a + 5)
3. (x³ − 2x + 1) − (x³ + x² − 2x)
4. Subtract 3y² − y from y² + 4y.
5. A rectangle has sides 3x + 2 and x − 1. Find its perimeter.

Answers: 1) 7x² + 2x − 4 2) 5a − 8 3) −x² + 1 4) −2y² + 5y 5) 8x + 2

Quick recap

• Add polynomials by combining like terms.
• Subtract by adding the opposite of every term.
• Write answers in standard form.