Adding and Subtracting Polynomials

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

Advanced25 minLesson
Printable worksheet

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:
  • 3x23x^2 and 5x25x^2 are like terms (both have x2x^2)
  • 2x2x and −7x-7x are like terms (both have x1x^1)
  • 44 and −9-9 are like terms (both are constants)
Unlike terms cannot be combined:
  • 3x23x^2 and 5x5x are NOT like terms (different powers)
  • 2x2x and 2y2y are NOT like terms (different variables)
To add polynomials: Combine like terms by adding their coefficients.
(3x2+2x+1)+(5x2−4x+3)=(3x2+5x2)+(2x−4x)+(1+3)=8x2−2x+4\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.
(3x2+2x+1)−(5x2−4x+3)=3x2+2x+1−5x2+4x−3=−2x2+6x−2\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: (4x2+3x−5)+(2x2−7x+8)(4x^2 + 3x - 5) + (2x^2 - 7x + 8)
1

Identify like terms

x2x^2 terms: 4x24x^2 and 2x22x^2 xx terms: 3x3x and −7x-7x Constants: −5-5 and 88 → Three groups of like terms
2

Add the x2x^2 terms

4x2+2x2=6x24x^2 + 2x^2 = 6x^2
3

Add the xx terms

3x+(−7x)=3x−7x=−4x3x + (-7x) = 3x - 7x = -4x
4

Add the constants

−5+8=3-5 + 8 = 3
5

Write the final polynomial

Combine all results in standard form → 6x2−4x+36x^2 - 4x + 3

Common Mistakes

Forgetting to distribute the negative sign to ALL terms when subtracting

Why it's wrong: When subtracting (3x2+2x−1)(3x^2 + 2x - 1), the minus applies to every term inside the parentheses.

Correct: (5x2)−(3x2+2x−1)=5x2−3x2−2x+1(5x^2) - (3x^2 + 2x - 1) = 5x^2 - 3x^2 - 2x + 1, NOT 5x2−3x2+2x−15x^2 - 3x^2 + 2x - 1

Adding unlike terms together

Why it's wrong: 3x2+5x3x^2 + 5x cannot be simplified because x2x^2 and xx are different powers.

Correct: Only combine terms with the SAME variable AND the SAME exponent. 3x2+5x3x^2 + 5x is already simplified.

Confusing coefficients with exponents when combining

Why it's wrong: When adding 3x2+5x23x^2 + 5x^2, you add the coefficients (3 and 5), not the exponents.

Correct: 3x2+5x2=8x23x^2 + 5x^2 = 8x^2 (NOT 8x48x^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: (x3+2x)+(5x2−x)=x3+5x2+x(x^3 + 2x) + (5x^2 - x) = x^3 + 5x^2 + x. The x3x^3 and 5x25x^2 have no partners.

Interactive Visual

Algebra tiles: combining like terms

Each tile is a term: big squares are x², long strips are x, small squares are 1. Red tiles are negative.

4x24x^2
3x3x
−5-5
2x22x^2
−7x-7x
88

4x2+3x−5+2x2−7x+84x^2 + 3x - 5 + 2x^2 - 7x + 8

Every term becomes tiles: the coefficient tells how many.

=x2= x^2=x= x=1= 1negative

Function graph

-3-2-10123-10010203040xy

Tap or hover over the graph to read off values.

p(x)q(x)p(x) + q(x)

Table of values

x-2-1012
p(x)5-4-5217
q(x)3017832
p(x) + q(x)35133519

Interactive Sandbox

Expression Calculator

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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+12at2s = 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)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−0.5x2R(x) = 50x - 0.5x^2 and cost is C(x)=20x+100C(x) = 20x + 100, then profit P(x)=R(x)−C(x)=50x−0.5x2−20x−100=−0.5x2+30x−100P(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−2x2R(x) = 100x - 2x^2 and cost C(x)=30x+500C(x) = 30x + 500 for producing xx items.

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

Step 1: Write the mathematical expression

Subtract: (100x−2x2)−(30x+500)(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 d1=3t2+2td_1 = 3t^2 + 2t and d2=t2−5t+10d_2 = t^2 - 5t + 10, the total displacement is d=d1+d2=4t2−3t+10d = d_1 + d_2 = 4t^2 - 3t + 10

2Try It Yourself

Two components of motion are described by s1=5t2+3ts_1 = 5t^2 + 3t and s2=2t2−8t+4s_2 = 2t^2 - 8t + 4.

Find the total motion s=s1+s2s = s_1 + s_2.

Step 1: Write the mathematical expression

Add: (5t2+3t)+(2t2−8t+4)(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 A1=x2+6xA_1 = x^2 + 6x has an alcove with area A2=2x+5A_2 = 2x + 5 added. Total area: A1+A2=x2+8x+5A_1 + A_2 = x^2 + 8x + 5

3Try It Yourself

A rectangular plot has area A=x2+10x+24A = x^2 + 10x + 24. A shed with area S=2x+8S = 2x + 8 is built on it.

What is the remaining usable area?

Step 1: Write the mathematical expression

Subtract: (x2+10x+24)−(2x+8)(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. 3x2+5x3x^2 + 5x is already fully simplified.
No! These are unlike terms because they have different exponents. 3x2+5x3x^2 + 5x is already fully simplified.
The exponents stay the same! When adding 3x2+5x23x^2 + 5x^2, you only add the coefficients: 3+5=83 + 5 = 8, so the result is 8x28x^2.
Subtracting a polynomial means subtracting every term in it. The negative sign in front of the parentheses applies to all terms inside, so each sign must change.

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., 3x23x^2 and −5x2-5x^2)
Coefficient
The numerical factor in front of a variable (e.g., in 7x27x^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
Today's challengeOne problem, three tries

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