Adding and Subtracting Polynomials Calculator

Add or subtract P(x) and Q(x) with degrees 0 through 6, combine like terms automatically, and read R(x) in standard form. Built for algebra class, exam review, engineering math checks, and symbolic expression practice.

Read the full guide Polynomial calculator

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Polynomial calculator

Set each degree from 0 to 6, then type coefficients from the highest power down to the constant. Degree 0 is a single constant. Leave a cell blank to treat it as zero.

Operation

Add P(x) + Q(x) Subtract P(x) − Q(x)

P(x)

Degree

Enter coefficient values

Type a number for each term shown (highest power first). Blank means 0.

Term Enter value

x²

x³

x⁴

x⁵

x⁶

Q(x)

Degree

Enter coefficient values

Type a number for each term shown (highest power first). Blank means 0.

Term Enter value

x²

x³

x⁴

x⁵

x⁶

Result

P(x) =

Q(x) =

Enter at least one coefficient to see a non-zero polynomial.

Like terms are combined automatically. The result uses the variable x.

Using this calculator

Pick Add or Subtract, then set the degree for P(x) and Q(x) (0 through 6).
Type the coefficient for each power of x, from x^n down to the constant term.
The result R(x) updates as you type. Nothing is sent to a server.

What Are Polynomials?

A polynomial is an algebraic expression built from terms that are constants or constants multiplied by whole-number powers of a variable. In this calculator the variable is x. Each term has a coefficient (the number in front) and a power (the exponent on x). The highest power that appears with a nonzero coefficient is the degree.

Polynomial expressions appear whenever you model quantities that change in smooth steps: revenue curves, motion approximations, area formulas, and exam drill problems all use the same structure. Adding and subtracting polynomials means you combine two expressions that use the same variable by matching powers and operating on coefficients only.

Terms, coefficients, and variables work together: in 5x^3 − 2x + 7 the coefficient of x^3 is 5, the coefficient of x is −2, and the constant term is 7. You never add exponents during polynomial addition or subtraction. That rule belongs to multiplication.

Monomial
One term, such as 4x^2 or −9.
Binomial
Two terms, such as x + 3 or 2x^2 − 5.
Trinomial
Three terms, such as x^2 + x − 1.
Real-world use
Sum or difference of cost, position, or data-fit curves that share the same variable.

Rules for Adding and Subtracting Polynomials

Polynomial addition and subtraction follow one core rule: combine like terms. Like terms share the same variable and the same power. You add or subtract the coefficients and keep the power unchanged.

Handle positive and negative signs carefully, especially in subtraction. Distribute the minus sign to every term of the second polynomial before you combine. Missing powers mean coefficient 0, not “skip this row.”

Let P(x) = a_n x^n + … + a_1 x + a_0 and Q(x) = b_m x^m + … + b_1 x + b_0.

Addition: (P + Q)(x) = Σ (a_k + b_k) x^k for each power k from 0 to max(n, m).

Subtraction: (P − Q)(x) = Σ (a_k − b_k) x^k for each power k from 0 to max(n, m).

Like terms: same variable, same exponent. Coefficients combine; exponents do not.

After combining, write the answer in standard form (descending powers) and drop any term whose coefficient is 0 unless the entire expression is 0. The calculator above applies these rules instantly for degrees 0 through 6.

How to Add and Subtract Polynomials

Whether you add or subtract, start by writing each polynomial in standard form with terms in descending power order. Align like terms, then combine coefficients column by column. Subtraction is safer when you first rewrite Q(x) as −Q(x) and add.

You can work horizontally in one line, vertically in columns, or directly in the calculator fields; the math is the same. Choose the layout that helps you avoid sign errors.

Horizontal method

Write P(x) + Q(x) or P(x) − Q(x), group like terms, and simplify. Good for short binomials and trinomials.

Vertical method

Stack terms by power in columns, add or subtract coefficients downward. Helpful when degrees differ or signs are messy.

Calculator method

Enter coefficients for P(x) and Q(x), choose Add or Subtract, and read R(x). Best for quick checks and higher-degree practice.

Adding Polynomials Examples

These examples show monomial, binomial, and trinomial addition. Enter the same coefficients in the calculator with operation Add to verify.

Monomial + monomial

P(x) = 3x^2 and Q(x) = −x^2.

Align: Both terms are x^2 (like terms).
Combine: 3 + (−1) = 2.

Result: P(x) + Q(x) = 2x^2.

Binomial + binomial

P(x) = 2x^2 + 3x + 1 and Q(x) = x^2 − x + 4.

x^2: 2 + 1 = 3.
x: 3 + (−1) = 2.
Constant: 1 + 4 = 5.

Result: P(x) + Q(x) = 3x^2 + 2x + 5.

Trinomial + trinomial (different degrees)

P(x) = 5x^4 − 2x and Q(x) = −x^2 + 7.

Missing terms: Treat Q as 0x^4 and 0x^3.
Combine: x^4: 5; x^2: −1; x: −2; constant: 7.

Result: P(x) + Q(x) = 5x^4 − x^2 − 2x + 7.

Subtracting Polynomials Examples

Subtraction requires distributing the minus sign to every term of Q(x). These walks show sign handling and simplification.

Subtract a binomial

P(x) = 7x − 4 and Q(x) = 2x + 1.

Rewrite: P(x) − Q(x) = (7x − 4) − (2x + 1).
Distribute: 7x − 4 − 2x − 1.
Combine: 5x − 5.

Result: P(x) − Q(x) = 5x − 5.

Subtract when degrees match

P(x) = 4x^3 + x − 2 and Q(x) = x^3 + 2x^2 + x + 1.

x^3: 4 − 1 = 3.
x^2: 0 − 2 = −2.
x: 1 − 1 = 0 (omit).
Constant: −2 − 1 = −3.

Result: P(x) − Q(x) = 3x^3 − 2x^2 − 3.

Subtract constants (degree 0)

P(x) = 12 and Q(x) = −5 (both degree 0).

Combine: 12 − (−5) = 17.

Result: P(x) − Q(x) = 17.

Combining Like Terms

Like terms are terms with the same variable raised to the same power. In polynomial addition and subtraction, combining like terms is the entire simplification step: you never change exponents, only coefficients.

Variable powers must match exactly: x^2 and x^3 are not like terms. Coefficient operations use ordinary real-number addition or subtraction, including negatives and decimals.

3x^2 and 5x^2 combine to 8x^2.
x and 4x combine to 5x.
Constants combine with constants only.
A missing x^2 term means coefficient 0 for x^2, not “no x^2 exists in the problem.”

Polynomial Operations

This page focuses on addition and subtraction. Other operations build on the same standard-form vocabulary.

Addition
Add coefficients of matching powers; degree of the sum is at most the larger input degree.
Subtraction
Subtract coefficients after negating Q(x), or use P + (−Q).
Multiplication (overview)
Multiply each term of one polynomial by every term of the other; exponents add. Not handled by this tool.
Division (overview)
Long or synthetic division reduces degree; different rules than add/subtract.
Identities
Results stay polynomials when you only add or subtract, with no fractions unless coefficients are fractions.

Simplifying Polynomial Expressions

After you add or subtract, simplification means writing the polynomial in standard form, combining all like terms, and removing zero coefficients. The calculator displays that cleaned form as R(x).

Expression evaluation at a number x is separate: first simplify the polynomial, then substitute. For add/subtract drills, focus on coefficient alignment before any substitution.

Write terms in descending power order.
Combine like terms once; do not mix powers.
Use parentheses when subtracting so every sign in Q(x) flips.
Check work with the calculator using the same degrees and coefficients.

Common Polynomial Mistakes

Most errors come from misaligned terms or mishandled signs, not from hard arithmetic. Slow down on subtraction and on missing middle terms.

Subtracting without distributing the minus sign to every term of Q(x).
Adding coefficients of different powers (for example mixing x^2 with x^3).
Adding exponents when only coefficients should combine.
Forgetting that a missing term means coefficient 0.
Dropping the constant term when only one polynomial has a constant.

Polynomial Addition vs Polynomial Subtraction

Both operations align like terms; subtraction adds the extra step of sign change on the second polynomial.

Core action

Addition: Add coefficients of each matching power.

Subtraction: Subtract coefficients, or add the opposite of Q(x).

Sign handling

Addition: Keep signs as written in both polynomials.

Subtraction: Flip every sign in Q(x) if you rewrite as addition.

Formula

Addition: (P + Q)(x) = Σ (a_k + b_k) x^k

Subtraction: (P − Q)(x) = Σ (a_k − b_k) x^k

Order

Addition: P + Q equals Q + P.

Subtraction: P − Q is generally not equal to Q − P.

Calculator use

Addition: Choose Add and enter both polynomials.

Subtraction: Choose Subtract; the tool negates Q(x) for you.

FAQs About Polynomial Operations

What degrees are supported?

Each of P(x) and Q(x) can have degree 0 through 6. Degree 0 is a constant. Higher degrees show every power down to the constant.

How do I subtract polynomials?

Choose Subtract, enter Q(x), and the calculator negates each coefficient before combining. On paper, rewrite as P(x) + (−Q(x)).

What are like terms?

Terms with the same variable and the same exponent. Only the coefficients add or subtract.

What if I leave a coefficient blank?

A blank cell is treated as zero, which is useful when a polynomial has no x^2 term, for example.

Does the calculator multiply polynomials?

No. This tool adds and subtracts only. Multiplication and factoring need different methods.

Can I use negative or decimal coefficients?

Yes. Type negatives and decimals as in a notebook. Results round for display when needed.

Are my entries stored online?

No. Calculations run in your browser; coefficients are not uploaded.

Why does the result omit some powers?

Terms with coefficient zero are hidden so the answer matches standard simplified form.