Operations on Polynomials

Key Terms

Adding Polynomials

Subtracting Polynomials

Multiplying Polynomials

Special Products

FOIL Method

Polynomial Long Division

Synthetic Division

Degree and Operations

Summary: The Four Operations at a Glance

Polynomial Operations FAQ

The Arithmetic of Polynomials

Numbers have addition, subtraction, multiplication, and division — and so do polynomials. The same four operations that govern ordinary arithmetic extend naturally to polynomial expressions, each with its own set of rules governing how terms interact, how degrees change, and what form the result takes.

Some of these operations are simple: adding polynomials amounts to collecting like terms. Others demand more care — polynomial long division mirrors its numerical counterpart but requires tracking multiple terms across multiple steps. Together, these operations provide the machinery behind factoring, solving polynomial equations, and simplifying complex algebraic expressions.

Key Terms

Addition and Subtraction

Like Terms— terms sharing the same variable and exponent; the basis of addition and subtraction

Term (of a Polynomial)— each unit has a coefficient and a degree

Multiplication

Degree (of a Polynomial)— degree of a product equals the sum of the input degrees

Binomial— multiplying two binomials is the most common multiplication task

Division

Polynomial Long Division— systematic division producing quotient and remainder

Synthetic Division— shortcut for dividing by

(x - c)

using only coefficients

Remainder Theorem— remainder equals

P(c)

Formulas Used on This Page

Square of a Sum—

(a + b)^2 = a^2 + 2ab + b^2

Square of a Difference—

(a - b)^2 = a^2 - 2ab + b^2

Cube of a Sum—

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Cube of a Difference—

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

See All Algebra Definitions →

See All Algebra Formulas →

Adding Polynomials

Adding two polynomials reduces to a single principle: combine like terms. Two terms are like terms when they share the same variable raised to the same exponent —

3x^2

and

5x^2

are like terms, while

3x^2

and

5x^3

are not.

Given

(2x^2 + 3x - 1) + (x^2 - 5x + 4)

, group the terms by degree: the

x^2

terms are

2x^2

and

x^2

, the

x

terms are

3x

and

-5x

, and the constants are

-1

and

4

. Adding each group gives

3x^2 - 2x + 3

.

The process works identically for any number of terms and any degree. Align terms by matching exponents, add their coefficients, and write the result. Terms with no counterpart in the other polynomial pass through unchanged — adding

x^3 + 2x

and

3x^2 - x + 5

produces

x^3 + 3x^2 + x + 5

, where

x^3

and

3x^2

each carried over directly.

The sum of two polynomials is always a polynomial. Its degree is at most the larger of the two input degrees — and in rare cases it can be less, if the leading terms cancel. Adding

2x^3 + x - 1

and

-2x^3 + 4x + 7

eliminates the cubic terms entirely, leaving

5x + 6

, a polynomial of degree

1

Subtracting Polynomials

Subtraction follows the same logic as addition, with one extra step: distribute the negative sign across every term of the polynomial being subtracted before combining like terms.

For

(2x^2 + 3x - 1) - (x^2 - 5x + 4)

, first negate the second polynomial:

-x^2 + 5x - 4

. Then add:

(2x^2 + 3x - 1) + (-x^2 + 5x - 4) = x^2 + 8x - 5

. Every term inside the parentheses changes sign — including terms that were already negative. The

-5x

becomes

+5x

and the

+4

becomes

-4

.

This is where most errors occur. Negating only the first term of the subtracted polynomial while leaving the rest untouched is the single most common mistake in polynomial subtraction. The expression

(5x^2 + 2x - 3) - (x^2 - 4x + 1)

is sometimes incorrectly computed as

4x^2 - 2x - 2

when only

x^2

gets negated. The correct result is

4x^2 + 6x - 4

.

Like addition, subtraction always produces a polynomial. The degree of the result is at most the larger of the two input degrees, with the same possibility of cancellation reducing it further.

Multiplying Polynomials

Multiplication requires every term of one polynomial to be multiplied by every term of the other. For each pair, multiply the coefficients and add the exponents on matching variables, then combine like terms in the resulting expression.

For

(x + 2)(x^2 - 3x + 1)

, distribute each term of the binomial across the trinomial. The term

x

produces

x \cdot x^2 = x^3

x \cdot (-3x) = -3x^2

, and

x \cdot 1 = x

. The term

2

produces

2 \cdot x^2 = 2x^2

2 \cdot (-3x) = -6x

, and

2 \cdot 1 = 2

. Collecting everything:

x^3 - 3x^2 + x + 2x^2 - 6x + 2

. Combining like terms gives

x^3 - x^2 - 5x + 2

.

The degree of a product equals the sum of the degrees of the factors — always, without exception. A degree-

2

polynomial times a degree-

3

polynomial produces a degree-

5

polynomial. This follows from the fact that the highest-degree terms in each factor multiply to produce the highest-degree term in the product, and no other pair of terms can reach that degree.

The number of terms before combining can grow quickly. Multiplying an

m

-term polynomial by an

n

-term polynomial generates

m \times n

individual products. A binomial times a trinomial produces

6

terms before simplification; two four-term polynomials produce

16

. Careful bookkeeping — and a systematic approach to distributing — prevents missed terms.

(x² − 3x + 2)(2x + 5): six cells, four buckets

Three terms times two terms fill a 3 × 2 grid of six products, and the like-term buckets sort them by degree: 2x³, then 5x² − 6x², then −15x + 4x, then 10. Multiplying coefficients and adding exponents happens once per cell; combining like terms happens once per bucket. Fill a grid of any size on the polynomial multiplication visualizer.

Long division, treated next, runs this grid backwards one cell at a time.

Special Products

Certain multiplication patterns appear so frequently that recognizing them on sight saves significant work. Each is a direct consequence of distribution, but memorizing the results avoids repeating the same steps every time.

Squaring a binomial produces a trinomial:

Square of a Sum

(a + b)^2 = a^2 + 2ab + b^2

Learn more about this formula: Square of a Sum →

Square of a Difference

(a - b)^2 = a^2 - 2ab + b^2

Learn more about this formula: Square of a Difference →

Browse all algebra formulas →

The middle term is always twice the product of the two original terms — a detail that connects directly to identifying perfect square trinomials when factoring.

Cubing a binomial follows the same pattern with four terms in the expansion:

Cube of a Sum

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Learn more about this formula: Cube of a Sum →

Cube of a Difference

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Learn more about this formula: Cube of a Difference →

Browse all algebra formulas →

The coefficients

1, 3, 3, 1

are the fourth row of Pascal's triangle — a pattern that generalizes through the binomial theorem covered in identities.

Multiplying conjugate binomials — a sum and a difference of the same two terms — eliminates the middle term entirely. The product

(a + b)(a - b)

equals

a^2 - b^2

, a clean difference of squares. No linear term survives because the outer and inner products cancel.

The cubic identities extend the same logic one degree further:

(a + b)(a^2 - ab + b^2) = a^3 + b^3

(a - b)(a^2 + ab + b^2) = a^3 - b^3

Each pairs a linear factor with a quadratic factor to produce a two-term cubic. These patterns are the multiplication side of the sum and difference of cubes formulas used in factoring — the same identities read in the opposite direction.

Pattern	Expansion
Square of a sum (a + b)²	a² + 2ab + b²
Square of a difference (a − b)²	a² − 2ab + b²
Conjugate binomials (a + b)(a − b)	a² − b² (difference of squares)
Sum of cubes (a + b)(a² − ab + b²)	a³ + b³
Difference of cubes (a − b)(a² + ab + b²)	a³ − b³

(x + 1)(x² − x + 1): the middle buckets cancel

The six cells deliver x³, −x² + x², x − x and 1, and the two middle buckets sum to zero, leaving x³ + 1: the sum-of-cubes pattern is a grid whose cross terms cancel. Every special product is a grid with a memorable cancellation. Watch the buckets empty on the polynomial multiplication visualizer.

Recognising the pattern saves filling the grid, but the grid is why the pattern holds.

FOIL Method

FOIL is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. It names the four products that arise when each term of one binomial is distributed across the other.

For

(2x + 3)(x - 4)

, the four steps produce: First

2x \cdot x = 2x^2

, Outer

2x \cdot (-4) = -8x

, Inner

3 \cdot x = 3x

, Last

3 \cdot (-4) = -12

. Combining the middle terms gives

2x^2 - 5x - 12

.

FOIL is nothing more than a structured way to ensure all four products are accounted for — it is a special case of the general distribution used in multiplying any two polynomials. Its value lies in speed and reliability for the specific case of two binomials, where exactly four products always arise.

The method does not extend beyond binomials. Attempting to apply FOIL to a binomial times a trinomial, or to two trinomials, misses terms. A binomial times a trinomial requires six products, not four. For anything beyond two binomials, the full term-by-term distribution described in the previous section is the correct approach.

First, Outer, Inner, Last: the four cells of (x + 2)(x + 3)

The mnemonic names the four cells of a 2 × 2 grid in a fixed order: First is the top-left x², Outer and Inner are the two x-cells that combine to 5x, Last is the constant 6. FOIL is the grid method for binomials, and it does not extend to longer factors, while the grid does. Compare FOIL with a 3 × 2 grid on the polynomial multiplication visualizer.

For anything larger than two binomials, return to the grid.

Polynomial Long Division

Dividing one polynomial by another follows a process that mirrors numerical long division. The idea is to find a quotient and a remainder that satisfy:

\text{dividend} = (\text{divisor} \times \text{quotient}) + \text{remainder}

To divide

x^3 - 2x^2 + 5x - 3

x - 1

, begin by dividing the leading term of the dividend by the leading term of the divisor:

x^3 \div x = x^2

. Multiply

x^2

by the entire divisor:

x^2(x - 1) = x^3 - x^2

. Subtract this from the dividend to get

-x^2 + 5x - 3

. Repeat:

-x^2 \div x = -x

, multiply

-x(x - 1) = -x^2 + x

, subtract to get

4x - 3

. One more pass:

4x \div x = 4

, multiply

4(x - 1) = 4x - 4

, subtract to get

1

. The quotient is

x^2 - x + 4

with remainder

1

.

The process terminates when the degree of what remains is less than the degree of the divisor. If the remainder is zero, the divisor divides the dividend evenly — meaning the divisor is a factor of the dividend.

Missing powers in the dividend require placeholder terms. Dividing

x^3 + 4

x + 1

requires writing the dividend as

x^3 + 0x^2 + 0x + 4

to keep terms aligned through each subtraction step.

The remainder on the graph

The section's division of x³ − 2x² + 5x − 3 by x − 1 gives quotient x² − x + 4 and remainder 1. The dashed curve is divisor times quotient, (x − 1)(x² − x + 4); the solid curve is the dividend. They differ by exactly 1 at every x, and at x = 1, where the divisor is zero, the dividend's value P(1) = 1 is the remainder itself.

The remainder theorem is this picture: dividing by x − c leaves P(c).

Synthetic Division

When the divisor is a linear binomial of the form

(x - c)

, synthetic division offers a streamlined alternative to long division. It uses only the coefficients of the dividend and the value

c

, reducing the process to a compact sequence of multiplications and additions.

To divide

x^3 - 2x^2 + 5x - 3

(x - 1)

, write the coefficients

1, -2, 5, -3

in a row and place

c = 1

to the left. Bring down the first coefficient:

1

. Multiply by

c

1 \cdot 1 = 1

. Add to the next coefficient:

-2 + 1 = -1

. Multiply by

c

-1 \cdot 1 = -1

. Add:

5 + (-1) = 4

. Multiply:

4 \cdot 1 = 4

. Add:

-3 + 4 = 1

. The bottom row reads

1, -1, 4, 1

— giving a quotient of

x^2 - x + 4

with remainder

1

, matching the long division result exactly.

The connection to polynomial evaluation is immediate. The remainder produced by synthetic division equals

P(c)

, the value of the dividend evaluated at

x = c

. This is the Remainder Theorem in action, and it means synthetic division doubles as a fast way to compute

P(c)

without substituting into every term. If

P(c) = 0

, then

(x - c)

divides the polynomial evenly and

c

is a root.

The limitation is strict: synthetic division works only for divisors of the form

(x - c)

. Dividing by

x^2 + 1

, by

2x - 3

, or by any polynomial of degree two or higher requires the full long division procedure.

Method	Divisor allowed	Process	Side benefit
Polynomial long division	any polynomial divisor	divide leading term, multiply, subtract; repeat until remainder degree < divisor degree	universal — works on every division problem
Synthetic division	linear divisors of the form (x − c) only	bring down, multiply by c, add — using only coefficients	remainder equals P(c); doubles as evaluation and root testing

Degree and Operations

Each arithmetic operation affects the degree of the result in a specific, predictable way. These rules serve as both a guide during computation and a check after it.

For addition and subtraction, the degree of the result is at most the maximum of the two input degrees. If

P(x)

has degree

4

and

Q(x)

has degree

3

, then

P(x) + Q(x)

has degree at most

4

. The "at most" matters — when the leading terms cancel, the degree drops. Adding

2x^3 + x

and

-2x^3 + 5x^2

produces

5x^2 + x

, a degree-

2

result from two degree-

3

inputs.

For multiplication, the degree of the product equals the sum of the degrees of the factors, with no exceptions. A degree-

2

polynomial times a degree-

3

polynomial always produces a degree-

5

polynomial. Leading coefficients multiply but never cancel, since neither is zero.

For division, the degree of the quotient equals the degree of the dividend minus the degree of the divisor. Dividing a degree-

5

polynomial by a degree-

2

polynomial produces a degree-

3

quotient. The remainder, if nonzero, has degree strictly less than the divisor.

These relationships provide a quick sanity check. If multiplying two quadratics yields a degree-

3

result, something went wrong. If dividing a cubic by a linear polynomial produces a cubic quotient, a step was missed.

When leading terms cancel

The section's 2x³ + x and −2x³ + 5x² are both cubics, but their sum is 5x² + x, a parabola: the leading terms 2x³ and −2x³ cancel. At x = 1 the bars show the sum point by point, 3 + 3 = 6. This is why the degree of a sum is only at most the larger degree.

Products never lose degree this way: the leading terms multiply and cannot cancel.

Summary: The Four Operations at a Glance

The four operations on polynomials — addition, subtraction, multiplication, and division — each follow a distinct procedure and affect the degree of the result in a predictable way. The table below collects every operation with its core technique, its degree effect, and the typical pitfall to watch for.

Operation	Core technique	Effect on degree	Watch out for
Addition	combine like terms	at most max(deg P, deg Q); less if leading terms cancel	aligning terms by matching exponent
Subtraction	distribute the negative across every term, then combine	same as addition	negating ONLY the first term — the classic error
Multiplication	every term × every term, then combine like terms (FOIL for binomial × binomial)	exactly deg P + deg Q	missing one of m × n products in big multiplications
Division	long division (general) or synthetic division (linear divisors only)	deg quotient = deg P − deg Q; remainder has degree < deg Q	missing-power placeholders; remainder may be nonzero

Polynomial Operations FAQ

Why isn't $(5x^2 + 2x - 3) - (x^2 - 4x + 1)$ equal to $4x^2 - 2x - 2$ ?

Because the minus sign has to reach every term in the second bracket, not just the first. Negating fully gives

-x^2 + 4x - 1

, so the sum is

4x^2 + 6x - 4

. Terms that were already negative flip too: the

-4x

becomes

+4x

. Negating only the leading term is the single most common error in polynomial subtraction.Read more →

Can adding two cubics produce a quadratic?

Yes, when the leading terms cancel. Adding

2x^3 + x

-2x^3 + 5x^2

gives

5x^2 + x

, a degree-two result from two degree-three inputs. That is why the rule for sums says the degree is at most the larger input degree. Multiplication never behaves this way: degrees add exactly, since neither leading coefficient is zero.Read more →

How do you handle missing terms in polynomial long division?

Write them in with zero coefficients so the columns line up. Dividing

x^3 + 4

x + 1

means treating the dividend as

x^3 + 0x^2 + 0x + 4

. A missing power is a zero coefficient rather than an absent slot, and leaving the gap open causes terms to misalign during the subtraction steps and corrupts the quotient.Read more →

Operations on Polynomials

The Arithmetic of Polynomials

Key Terms

Addition and Subtraction

Multiplication

Division

Formulas Used on This Page

Adding Polynomials

Subtracting Polynomials

Multiplying Polynomials

Special Products

FOIL Method

Polynomial Long Division

Synthetic Division

Degree and Operations

Summary: The Four Operations at a Glance

Polynomial Operations FAQ

Why isn't (5x2+2x−3)−(x2−4x+1)(5x^2 + 2x - 3) - (x^2 - 4x + 1)(5x2+2x−3)−(x2−4x+1) equal to 4x2−2x−24x^2 - 2x - 24x2−2x−2?

Can adding two cubics produce a quadratic?

How do you handle missing terms in polynomial long division?

Why isn't $(5x^2 + 2x - 3) - (x^2 - 4x + 1)$ equal to $4x^2 - 2x - 2$ ?