Adding and Subtracting Polynomials

Learn how to add and subtract polynomials by combining like terms.

Advanced25 minLesson

Definition

Adding and subtracting polynomials means combining like terms - terms that have the same variable raised to the same power.

Like terms share the same variable and exponent:

$3x^2$ and $5x^2$ are like terms (both have $x^2$ )
$2x$ and $-7x$ are like terms (both have $x^1$ )
$4$ and $-9$ are like terms (both are constants)

Unlike terms cannot be combined:

$3x^2$ and $5x$ are NOT like terms (different powers)
$2x$ and $2y$ are NOT like terms (different variables)

To add polynomials: Combine like terms by adding their coefficients.

\begin{align}(3x^2 + 2x + 1) + (5x^2 - 4x + 3) &= (3x^2 + 5x^2) + (2x - 4x) + (1 + 3) \\ &= 8x^2 - 2x + 4\end{align}

To subtract polynomials: Distribute the negative sign, then combine like terms.

\begin{align}(3x^2 + 2x + 1) - (5x^2 - 4x + 3) &= 3x^2 + 2x + 1 - 5x^2 + 4x - 3 \\ &= -2x^2 + 6x - 2\end{align}

Try it now

Which terms are like terms?

Worked Examples

Add: $(4x^2 + 3x - 5) + (2x^2 - 7x + 8)$

Identify like terms

$x^2$ terms: $4x^2$ and $2x^2$ $x$ terms: $3x$ and $-7x$ Constants: $-5$ and $8$ → Three groups of like terms

Add the $x^2$ terms

$4x^2 + 2x^2 = 6x^2$

Add the $x$ terms

$3x + (-7x) = 3x - 7x = -4x$

Add the constants

$-5 + 8 = 3$

Write the final polynomial

Combine all results in standard form → $6x^2 - 4x + 3$

Answer: $6x^2 - 4x + 3$

Common Mistakes

Forgetting to distribute the negative sign to ALL terms when subtracting

Why it's wrong: When subtracting $(3x^2 + 2x - 1)$ , the minus applies to every term inside the parentheses.

Correct: $(5x^2) - (3x^2 + 2x - 1) = 5x^2 - 3x^2 - 2x + 1$ , NOT $5x^2 - 3x^2 + 2x - 1$

Adding unlike terms together

Why it's wrong: $3x^2 + 5x$ cannot be simplified because $x^2$ and $x$ are different powers.

Correct: Only combine terms with the SAME variable AND the SAME exponent. $3x^2 + 5x$ is already simplified.

Confusing coefficients with exponents when combining

Why it's wrong: When adding $3x^2 + 5x^2$ , you add the coefficients (3 and 5), not the exponents.

Correct: $3x^2 + 5x^2 = 8x^2$ (NOT $8x^4$ ). 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: $(x^3 + 2x) + (5x^2 - x) = x^3 + 5x^2 + x$ . The $x^3$ and $5x^2$ have no partners.

Interactive Visual

Balance Scale

x + 3=7

Apply to both sides:

Solution: x = 4

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

16 problems

Problem 1 of 16

Easy

Which terms are like terms?

Why It Matters

Polynomial operations are fundamental building blocks in algebra and beyond:

Physics: Combining motion equations like $s = ut + \frac{1}{2}at^2$ requires polynomial arithmetic
Engineering: Calculating areas and volumes often involves adding polynomial expressions
Economics: Profit functions combine revenue and cost polynomials: $P(x) = R(x) - C(x)$
Computer Science: Polynomial algorithms and complexity analysis use these operations

Mastering polynomial addition and subtraction prepares you for factoring, solving equations, and calculus!

Real World Applications

Business: Profit Calculations

Companies calculate profit by subtracting cost from revenue, where both are often polynomial functions.

Example:

If revenue is $R(x) = 50x - 0.5x^2$ and cost is $C(x) = 20x + 100$ , then profit $P(x) = R(x) - C(x) = 50x - 0.5x^2 - 20x - 100 = -0.5x^2 + 30x - 100$

1Try It Yourself

A company has revenue $R(x) = 100x - 2x^2$ and cost $C(x) = 30x + 500$ for producing $x$ items.

What is the profit polynomial $P(x) = R(x) - C(x)$ ?

Step 1: Write the mathematical expression

Subtract: $(100x - 2x^2) - (30x + 500)$

Physics: Combining Motion Equations

When objects move in the same direction, their position equations can be added.

Example:

If two forces create displacements $d_1 = 3t^2 + 2t$ and $d_2 = t^2 - 5t + 10$ , the total displacement is $d = d_1 + d_2 = 4t^2 - 3t + 10$

2Try It Yourself

Two components of motion are described by $s_1 = 5t^2 + 3t$ and $s_2 = 2t^2 - 8t + 4$ .

Find the total motion $s = s_1 + s_2$ .

Step 1: Write the mathematical expression

Add: $(5t^2 + 3t) + (2t^2 - 8t + 4)$

Architecture: Area Calculations

When combining or removing sections of floor plans, architects add or subtract polynomial area expressions.

Example:

A room with area $A_1 = x^2 + 6x$ has an alcove with area $A_2 = 2x + 5$ added. Total area: $A_1 + A_2 = x^2 + 8x + 5$

3Try It Yourself

A rectangular plot has area $A = x^2 + 10x + 24$ . A shed with area $S = 2x + 8$ is built on it.

What is the remaining usable area?

Step 1: Write the mathematical expression

Subtract: $(x^2 + 10x + 24) - (2x + 8)$

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

No! These are unlike terms because they have different exponents.

3x^2 + 5x

is already fully simplified.

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., $3x^2$ and $-5x^2$ )
Coefficient: The numerical factor in front of a variable (e.g., in $7x^2$ , 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

Adding and Subtracting Polynomials

Definition

Worked Examples

Common Mistakes

Interactive Visual

Balance Scale

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Business: Profit Calculations

Physics: Combining Motion Equations

Architecture: Area Calculations

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Introduction to Polynomials

FOIL Method

Classifying Polynomials

Multiplying Polynomials