Adding and Subtracting Polynomials
Learn how to add and subtract polynomials by combining like terms.
Definition
- and are like terms (both have )
- and are like terms (both have )
- and are like terms (both are constants)
- and are NOT like terms (different powers)
- and are NOT like terms (different variables)
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Worked Examples
Add:
Identify like terms
terms: and terms: and Constants: and → Three groups of like terms
Add the terms
→
Add the terms
→
Add the constants
→
Write the final polynomial
Combine all results in standard form →
Answer:
Common Mistakes
Forgetting to distribute the negative sign to ALL terms when subtracting
Why it's wrong: When subtracting , the minus applies to every term inside the parentheses.
Correct: , NOT
Adding unlike terms together
Why it's wrong: cannot be simplified because and are different powers.
Correct: Only combine terms with the SAME variable AND the SAME exponent. is already simplified.
Confusing coefficients with exponents when combining
Why it's wrong: When adding , you add the coefficients (3 and 5), not the exponents.
Correct: (NOT ). The exponent stays the same!
Forgetting terms with no like term
Why it's wrong: If there is no matching term, the term stands alone in the answer.
Correct: . The and have no partners.
Interactive Visual
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsWhich terms are like terms?
Why It Matters
- Physics: Combining motion equations like requires polynomial arithmetic
- Engineering: Calculating areas and volumes often involves adding polynomial expressions
- Economics: Profit functions combine revenue and cost polynomials:
- Computer Science: Polynomial algorithms and complexity analysis use these operations
Real World Applications
Business: Profit Calculations
Companies calculate profit by subtracting cost from revenue, where both are often polynomial functions.
Example:
If revenue is and cost is , then profit
A company has revenue and cost for producing items.
What is the profit polynomial ?
Step 1: Write the mathematical expression
Subtract:
Physics: Combining Motion Equations
When objects move in the same direction, their position equations can be added.
Example:
If two forces create displacements and , the total displacement is
Two components of motion are described by and .
Find the total motion .
Step 1: Write the mathematical expression
Add:
Architecture: Area Calculations
When combining or removing sections of floor plans, architects add or subtract polynomial area expressions.
Example:
A room with area has an alcove with area added. Total area:
A rectangular plot has area . A shed with area is built on it.
What is the remaining usable area?
Step 1: Write the mathematical expression
Subtract:
Key Takeaways
- 1Like terms have the same variable raised to the same power
- 2To add polynomials, combine the coefficients of like terms
- 3To subtract polynomials, distribute the negative sign to every term, then combine like terms
- 4Terms without a matching like term remain unchanged in the result
- 5Always write your final answer in standard form (highest degree first)
Frequently Asked Questions
Glossary
- Polynomial
- An expression with one or more terms, where each term has a variable raised to a non-negative integer power
- Like terms
- Terms that have exactly the same variable raised to the same power (e.g., and )
- Coefficient
- The numerical factor in front of a variable (e.g., in , the coefficient is 7)
- Standard form
- A polynomial written with terms in order from highest degree to lowest
- Degree
- The highest exponent in a polynomial; for a term, it is the exponent of the variable