Adding and Subtracting Polynomials

What is a Polynomial?

A polynomial is an expression made up of variables, coefficients, and exponents combined using addition and subtraction.

Examples of polynomials:

• 3x + 5
• x² - 4x + 7
• 2x³ + 5x² - x + 8
• 7 (constant is a polynomial)

Not polynomials:

• 1/x (negative exponent: x⁻¹)
• √x (fractional exponent: x^(1/2))
• x² + 3/x

Polynomial Vocabulary

Terms: Parts separated by + or - signs Example: 3x² - 5x + 7 has three terms

Coefficient: Number multiplying the variable In 5x², the coefficient is 5

Degree: Highest exponent in the polynomial

• 3x² + 5x - 1 has degree 2
• x³ - 4x² + x has degree 3
• 7 has degree 0

Leading Coefficient: Coefficient of the highest degree term In 2x³ - 5x² + 3x - 1, the leading coefficient is 2

Constant Term: The term without a variable In x² + 3x + 5, the constant is 5

Types of Polynomials by Number of Terms

Monomial: One term

• Examples: 5x², -3x, 7

Binomial: Two terms

• Examples: x + 5, 3x² - 7

Trinomial: Three terms

• Examples: x² + 5x + 6, 2x² - 3x + 1

Polynomial: Four or more terms (or general term)

• Example: x³ + 2x² - 5x + 3

Types of Polynomials by Degree

Linear (Degree 1): 3x + 5

Quadratic (Degree 2): x² + 3x - 2

Cubic (Degree 3): 2x³ - x² + 4x + 1

Quartic (Degree 4): x⁴ - 3x² + 2

Standard Form

Polynomials in standard form are written with:

• Terms in descending order of degree (highest to lowest)
• Like terms combined

Examples:

Not standard: 5 + 3x - 2x² Standard: -2x² + 3x + 5

Not standard: x² + 3x² - 4 + x Standard: 4x² + x - 4

Like Terms

Like terms have the same variable(s) raised to the same power(s).

Like terms:

• 3x and 5x (same variable, same power)
• 2x² and -7x² (same variable, same power)
• 4xy and -xy (same variables, same powers)

NOT like terms:

• 3x and 3x² (different powers)
• 2x and 2y (different variables)
• 5x²y and 5xy² (different powers)

Combining Like Terms

Add or subtract the coefficients; keep the variable part the same.

Example 1: 3x + 5x = (3 + 5)x = 8x

Example 2: 7x² - 2x² = (7 - 2)x² = 5x²

Example 3: 4x + 2y - x + 5y = (4x - x) + (2y + 5y) = 3x + 7y

Example 4: 5x² + 3x - 2x² + 7x = (5x² - 2x²) + (3x + 7x) = 3x² + 10x

Adding Polynomials

Method 1: Horizontal (Combine Like Terms)

Add by grouping like terms together.

Example 1: Add (3x + 5) + (2x + 7)

Remove parentheses: 3x + 5 + 2x + 7 Group like terms: (3x + 2x) + (5 + 7) Combine: 5x + 12

Example 2: Add (x² + 3x - 4) + (2x² - x + 5)

Remove parentheses: x² + 3x - 4 + 2x² - x + 5 Group like terms: (x² + 2x²) + (3x - x) + (-4 + 5) Combine: 3x² + 2x + 1

Example 3: Add (4x² - 2x + 1) + (x² + 5x - 3)

= 4x² - 2x + 1 + x² + 5x - 3 = (4x² + x²) + (-2x + 5x) + (1 - 3) = 5x² + 3x - 2

Method 2: Vertical (Column Method)

Align like terms in columns and add.

Example: Add (3x² + 5x - 2) + (x² - 3x + 7)

Write aligned:

• First row: 3x² + 5x - 2
• Second row: x² - 3x + 7
• Add columns: 4x² + 2x + 5

Subtracting Polynomials

Key Idea: Distribute the negative sign (or multiply by -1) to every term in the second polynomial, then add.

Example 1: Subtract (2x + 5) - (x + 3)

Distribute negative: 2x + 5 - x - 3 Group like terms: (2x - x) + (5 - 3) Combine: x + 2

Example 2: Subtract (3x² + 2x - 1) - (x² - 4x + 5)

Distribute negative: 3x² + 2x - 1 - x² + 4x - 5 Note: -(x²) = -x², -(-4x) = +4x, -(5) = -5 Group: (3x² - x²) + (2x + 4x) + (-1 - 5) Combine: 2x² + 6x - 6

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

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

Vertical Method for Subtraction:

Example: (4x² + 3x - 5) - (2x² - x + 3)

Write the first polynomial, then change signs of second and add: First: 4x² + 3x - 5 Second (signs changed): -2x² + x - 3

Add them together: 2x² + 4x - 8

Important: Distributing the Negative Sign

Common Mistake: Forgetting to distribute negative to all terms!

Wrong: (3x - 5) - (2x - 4) = 3x - 5 - 2x - 4 = x - 9 ✗

Right: (3x - 5) - (2x - 4) = 3x - 5 - 2x + 4 = x - 1 ✓

The negative must change ALL signs in the parentheses!

Complex Examples

Example 1: Add three polynomials (2x² + x) + (3x² - 4x + 1) + (x² + 2x - 3)

= 2x² + x + 3x² - 4x + 1 + x² + 2x - 3 = (2x² + 3x² + x²) + (x - 4x + 2x) + (1 - 3) = 6x² - x - 2

Example 2: Multiple operations (5x² + 2x - 3) + (2x² - x + 1) - (3x² + 4x - 2)

First add: (5x² + 2x²) + (2x - x) + (-3 + 1) = 7x² + x - 2

Then subtract: 7x² + x - 2 - 3x² - 4x + 2 = (7x² - 3x²) + (x - 4x) + (-2 + 2) = 4x² - 3x

Example 3: With fractions (1/2 x² + 3x) + (1/4 x² - 2x + 5)

= (1/2 x² + 1/4 x²) + (3x - 2x) + 5 = (2/4 x² + 1/4 x²) + x + 5 = 3/4 x² + x + 5

Polynomials with Multiple Variables

Apply the same rules - combine only like terms.

Example 1: Add (3xy + 2x) + (5xy - 4x)

= (3xy + 5xy) + (2x - 4x) = 8xy - 2x

Example 2: Subtract (4x²y - 3xy + 2) - (x²y + xy - 5)

= 4x²y - 3xy + 2 - x²y - xy + 5 = (4x²y - x²y) + (-3xy - xy) + (2 + 5) = 3x²y - 4xy + 7

Example 3: Add (2a²b + 3ab² - ab) + (a²b - 2ab² + 4ab)

= (2a²b + a²b) + (3ab² - 2ab²) + (-ab + 4ab) = 3a²b + ab² + 3ab

Simplifying Expressions

Always write final answers in standard form.

Example 1: Simplify 5 + 2x - 3x² + x - 4

Combine like terms: -3x² + (2x + x) + (5 - 4) = -3x² + 3x + 1

Example 2: Simplify 4x³ + 2x - x³ + 5x² - 3x + 1

= (4x³ - x³) + 5x² + (2x - 3x) + 1 = 3x³ + 5x² - x + 1

Evaluating Polynomials

After adding/subtracting, you may need to evaluate for a specific value.

Example: If x = 2, evaluate (3x² + 5x) + (x² - 2x + 3)

First simplify: 4x² + 3x + 3

Then substitute x = 2: = 4(2)² + 3(2) + 3 = 4(4) + 6 + 3 = 16 + 6 + 3 = 25

Perimeter Applications

Adding polynomials often appears in geometry problems.

Example: A rectangle has length (3x + 5) and width (2x - 1). Find the perimeter.

Perimeter = 2(length) + 2(width) = 2(3x + 5) + 2(2x - 1) = 6x + 10 + 4x - 2 = 10x + 8

Example 2: A triangle has sides (x + 3), (2x - 1), and (x + 5). Find the perimeter.

P = (x + 3) + (2x - 1) + (x + 5) = x + 3 + 2x - 1 + x + 5 = 4x + 7

Word Problems

Example: The cost to produce x items is (50x + 200) dollars. The revenue from selling x items is (80x - 50) dollars. What is the profit?

Profit = Revenue - Cost = (80x - 50) - (50x + 200) = 80x - 50 - 50x - 200 = 30x - 250

The profit is (30x - 250) dollars.

Common Mistakes to Avoid

1. Not distributing the negative sign (3x - 5) - (2x - 4) ≠ 3x - 5 - 2x - 4

2. Combining unlike terms 3x + 2x² ≠ 5x³ These cannot be combined!

3. Forgetting to write in standard form 5 + 3x - 2x² should be -2x² + 3x + 5

4. Sign errors with multiple operations Be extra careful when subtracting twice

5. Confusing coefficients and exponents 2x³ + 3x³ = 5x³, NOT 5x⁶

Checking Your Work

Method 1: Substitute a value Pick x = 1 and evaluate both the original expression and your answer. They should match.

Method 2: Use a different value Try x = 2 as well to be more confident.

Method 3: Reverse the operation For addition, subtract one polynomial from the sum to get the other.

Quick Reference

Practice Strategy

Level 1: Start with monomials and binomials

• 3x + 5x
• (2x + 3) + (x + 4)

Level 2: Move to trinomials

• (x² + 2x + 1) + (x² - x + 3)

Level 3: Practice subtraction carefully

• (3x² - 2x + 5) - (x² + 4x - 1)

Level 4: Multiple operations

• Add three or more polynomials
• Mix addition and subtraction

Level 5: Applications

• Perimeter problems
• Word problems
• Multiple variables

Tips for Success

• Write clearly and line up like terms
• Use parentheses when needed
• Check signs carefully, especially when subtracting
• Always simplify completely
• Write final answers in standard form
• Show all steps - don't skip!
• Practice, practice, practice!

Mental Math Shortcuts

For simple problems, combine mentally:

• (2x + 3) + (x + 5) = 3x + 8
• (5x - 2) - (3x + 1) = 2x - 3

For complex problems, write it out:

• Use horizontal or vertical method
• Show intermediate steps
• Double-check work