Polynomial Addition, Subtraction, and Multiplication

The Central Idea

A polynomial is a structured sum of power terms. Because each term is associated with a particular power of the variable, polynomial arithmetic can be organized around those powers.

Consider

and

Written with missing powers made explicit,

and

Addition and subtraction combine coefficients attached to the same power. Multiplication is broader: each term in one polynomial multiplies every term in the other.

Polynomials Are Closed Under These Operations

If \(P(x)\) and \(Q(x)\) are polynomials, then

and

are also polynomials.

We say that polynomials are closed under addition, subtraction, and multiplication.

These operations never require us to introduce negative powers, fractional powers, or variables in denominators. We remain inside the polynomial family.

Like Terms Are the Key to Addition and Subtraction

Two terms are like terms when they have the same variable part with the same exponents.

For example,

and

are like terms because both contain \(x^4\).

Their coefficients can be combined:

By contrast,

and

are not like terms. Their powers differ, so they cannot be combined by addition or subtraction.

Why Like Terms Combine

Combining like terms is an application of the distributive property.

Consider

Both terms contain the common factor \(x^2\), so

Therefore,

The power does not change because we are adding coefficients, not multiplying the powers.

Adding Polynomials

To add polynomials, combine all like terms.

Suppose

and

Then

is

Group like powers:

Combine the coefficients:

Therefore,

Addition as Coefficient Arithmetic

The previous example can be viewed entirely through coefficients.

The coefficient lists are

and

Add corresponding entries:

This gives

corresponding to

This coefficient-by-coefficient viewpoint is one reason standard form is so useful.

Adding Polynomials with Missing Powers

Missing powers represent zero coefficients.

Suppose

and

Exposing the missing powers gives

and

Add corresponding coefficients:

Vertical Addition

Polynomials may also be arranged vertically. Like powers should occupy the same column.

The vertical method is especially useful when polynomials contain many terms because the columns make the coefficient alignment visible.

Commutativity of Polynomial Addition

Because ordinary number addition is commutative, polynomial addition is also commutative:

For example,

and

both simplify to

Associativity of Polynomial Addition

Polynomial addition is also associative:

This means that when several polynomials are being added, we may regroup them without changing the result.

The familiar laws of arithmetic continue to operate on these larger algebraic objects.

Subtracting Polynomials

Polynomial subtraction is most reliable when we treat subtraction as addition of an additive inverse.

Suppose

and

We want

Write the subtraction explicitly:

Distribute the Negative Sign First

Subtracting the entire polynomial \(Q(x)\) means changing the sign of every term in \(Q(x)\).

Therefore,

Combine like terms:

Subtraction as Addition of the Additive Inverse

The same operation can be written

If

then its additive inverse is

Thus subtraction is not a completely separate operation. It is addition after replacing the second polynomial by its opposite.

Why Parentheses Matter in Subtraction

Consider

The negative sign applies to the entire second polynomial:

Then

A common mistake is to change only the sign of the first term in the second polynomial. Parentheses help show that every term is being subtracted.

Vertical Subtraction

Vertical subtraction works well when signs are handled carefully.

Suppose

First replace the second polynomial by its additive inverse:

Then add by columns:

Therefore,

Addition and Subtraction Do Not Usually Change Exponents

When like terms are added or subtracted, their coefficients change but their common power remains the same.

For example,

We do not add or subtract the exponents because the terms are not being multiplied or divided.

The expression

is

not \(x^4\).

The Degree of a Sum

If \(P\) and \(Q\) are nonzero polynomials, the degree of their sum cannot exceed the greater of their two degrees:

Frequently the degree equals that maximum, but leading terms can cancel.

For example,

simplifies to

The degree drops from \(4\) to \(1\) because the \(x^4\) terms cancel.

The Degree of a Difference

The same principle applies to subtraction:

provided the result is not the zero polynomial.

If

then

producing the zero polynomial, whose degree is left undefined in our current convention.

Multiplying Polynomials

Polynomial multiplication is built from two familiar ideas:

1. multiply coefficients;
2. add exponents when multiplying equal bases.

For monomials,

becomes

Therefore,

Multiplying Signed Monomials

The ordinary sign rules still apply.

Likewise,

The coefficient arithmetic and exponent arithmetic can be handled separately and then recombined.

Multiplying a Monomial by a Polynomial

To multiply a monomial by a polynomial, multiply the monomial by every term in the polynomial.

Consider

Apply the distributive property:

Multiply each pair:

Therefore,

Every Term Must Be Multiplied

In

the factor \(-2x\) multiplies all four terms:

This gives

Missing even one term changes the polynomial and makes the product incorrect.

Multiplying Two Polynomials

When both factors contain several terms, every term in the first polynomial must multiply every term in the second.

Consider

Multiply the first polynomial by \(x\):

Multiply the first polynomial by \(4\):

Add the partial products:

Combine like terms:

The Partial-Product Method

The previous multiplication can be organized vertically, much like multidigit multiplication.

Each row is a partial product. Once the rows are aligned by powers, like terms can be added.

This method scales well when the factors contain several terms.

A Term-by-Term View

The same product

can be expanded by listing all pairwise products:

This becomes

Then

The essential rule is simple: every term in one factor pairs with every term in the other.

Multiplying Two Binomials

Consider

Multiply \(2x\) by both terms in the second factor:

Multiply \(3\) by both terms:

Add:

Therefore,

Binomial multiplication has several useful shortcuts and recurring patterns. Those will be developed systematically in the next lesson on Distribution and Special Products.

Multiplying Polynomials with Missing Powers

Missing powers do not need to be inserted before ordinary multiplication, but keeping the degree structure visible can prevent errors.

Consider

Multiply \(x^3\) by every term:

Multiply \(2\) by every term:

Add:

Multiplying Longer Polynomials

Consider

Multiply by \(2x^2\):

Multiply by \(-3x\):

Multiply by \(1\):

Add the partial products:

Therefore,

Degree of a Product

For nonzero polynomials with real coefficients,

If

and

then the leading terms have the form

and

Their product is

Because \(a_m\neq0\) and \(b_n\neq0\), the coefficient \(a_mb_n\) is also nonzero, so the product has degree \(m+n\).

Example: Predict the Product Degree

Suppose

and

We know

and

Therefore, before multiplying anything else,

This prediction is a useful error check during expansion.

Leading Coefficient of a Product

The leading coefficient of a product is the product of the leading coefficients.

If

and

then

begins with

Thus the product has

and

A Fast Structural Check Before Expanding

Before multiplying two nonzero polynomials, inspect their leading terms.

For

the product must begin with

If a completed expansion has degree \(8\), degree \(10\), or a positive leading coefficient, something went wrong.

Structural predictions provide a quick independent check without repeating the entire calculation.

Constant Terms Also Multiply Predictably

The constant term of a product comes from multiplying the constant terms of the factors.

If

and

then the constant term of \(P(x)Q(x)\) is

For example, in

the constant term of the product must be

Why the Constant-Term Rule Works

Any product involving a positive power of \(x\) still contains a positive power of \(x\).

The only way to produce an \(x^0\) term from polynomial multiplication is therefore

Since

the resulting constant coefficient is simply

Polynomial Multiplication as Coefficient Convolution

The coefficient-list viewpoint can also describe multiplication.

Suppose

and

When the polynomials are multiplied, terms whose exponents add to the same value become like terms.

The constant coefficient is

The coefficient of \(x\) is

The coefficient of \(x^2\) is

This systematic combining of shifted coefficient products is called convolution.

The General Coefficient Rule for Multiplication

If

and

then

where

This compact formula says exactly what ordinary polynomial multiplication does: multiply coefficient pairs whose exponents add to \(k\), then add those products to obtain the coefficient of \(x^k\).

You do not need this notation to perform elementary multiplication, but it exposes the algorithm underneath the familiar procedure.

Example: See the Convolution

Let

and

The coefficient lists, written from \(x^0\) upward, are

and

The product coefficients are

and

Therefore,

Rewritten in descending standard form,

Why Computer Algebra Likes Coefficient Lists

A computer does not need to "see" a polynomial the way a person does. It can store a polynomial as a sequence of coefficients.

For example,

can be stored as

when coefficients are listed from highest degree to constant term.

Addition and subtraction then become element-by-element arithmetic. Multiplication becomes a systematic combination of coefficient products.

The same algebra learned by hand therefore connects directly to the algorithms used in symbolic and numerical computation.

Multiplication and Function Values

If

then evaluating \(R\) at any number \(a\) gives

This gives us another way to check a multiplication.

Suppose

At \(x=4\), the factored form gives

The expanded form gives

Agreement at a chosen test value does not by itself prove two formulas are identical, but it is an efficient way to catch many arithmetic errors.

Checking Addition by Evaluation

If

then

For example, let

and

Their sum is

At \(x=2\),

so

The simplified sum gives

Checking Subtraction by Evaluation

Similarly, if

then

This can be especially useful for detecting a missed sign when subtracting a polynomial.

Several Variables

The same operations work with multivariable polynomials.

For example,

becomes

so

The terms combine because \(x^2y\) matches \(x^2y\), and \(xy^2\) matches \(xy^2\).

Multiplication with Several Variables

When multiplying monomials, add exponents on matching variable bases.

gives

Therefore,

Each variable keeps its own exponent bookkeeping.

A Geometric Interpretation of Multiplication

Polynomial multiplication can represent area.

Suppose a rectangle has side lengths

and

Its area is

Dividing the rectangle according to the two parts of each side produces four smaller areas:

Adding them gives

Thus

The next lesson will develop this distributive structure more deeply and use it to recognize recurring special products.

Multiplication Is Commutative

Polynomial multiplication inherits the commutative property:

For example,

and

both produce

Multiplication Is Associative

Polynomial multiplication is associative:

This means a product of several polynomial factors may be grouped in whichever order is most convenient.

The grouping can change the amount of work, but not the final polynomial.

Multiplication Distributes Over Addition

Polynomials also obey the distributive property:

This law is the engine behind polynomial multiplication.

When we say that every term in one polynomial must multiply every term in another, we are repeatedly applying distribution.

The next lesson will focus specifically on this structure and the useful patterns it creates.

Common Error: Adding Unlike Terms

The expression

cannot be simplified to

The terms have different variable parts, so they are not like terms.

Unless another operation is being performed, the correct expression remains

Common Error: Adding Exponents During Addition

Exponents are added when powers with the same base are multiplied, not when like terms are added.

while

The operation determines which rule applies.

Common Error: Losing the Subtraction Sign

In

the entire second polynomial changes sign:

Therefore,

The safest habit is to rewrite subtraction as addition of the additive inverse before combining any terms.

Common Error: Multiplying Only the First Terms

For

multiplying only \(x\cdot x\) produces only one part of the product.

All four pairings are needed:

Therefore,

Common Error: Multiplying Coefficients but Forgetting Exponents

Consider

The coefficients multiply:

and the powers multiply:

Thus

Common Error: Failing to Combine Partial Products

An expansion is not finished merely because every multiplication has been performed.

For example,

The terms \(3x\) and \(2x\) are like terms, so standard form is

Multiplication creates partial products; simplification combines them.

A Reliable Procedure for Polynomial Addition

1. Write both polynomials in standard form.
2. Identify corresponding powers.
3. Treat missing powers as having coefficient \(0\).
4. Add the coefficients of like terms.
5. Write the result in standard form.
6. Check whether leading terms canceled and changed the degree.

A Reliable Procedure for Polynomial Subtraction

1. Place parentheses around the polynomial being subtracted.
2. Replace subtraction with addition of its additive inverse.
3. Change the sign of every term in the second polynomial.
4. Combine like terms.
5. Write the result in standard form.
6. Check signs and degree.

A Reliable Procedure for Polynomial Multiplication

1. Predict the degree from the degrees of the factors.
2. Predict the leading term from the leading terms of the factors.
3. Multiply every term in one factor by every term in the other.
4. Multiply coefficients and add exponents on equal variable bases.
5. Collect all partial products.
6. Combine like terms.
7. Write the result in standard form.
8. Compare the result with your predicted degree and leading term.

Example: A Complete Addition Workflow

Add

Align the powers:

Add corresponding coefficients:

Example: A Complete Subtraction Workflow

Subtract

Change every sign in the second polynomial:

Combine like terms:

Example: A Complete Multiplication Workflow

Multiply

First predict the leading term:

Now form the partial products:

Combine:

Therefore,

The result has degree \(4\) and leading term \(2x^4\), exactly as predicted.

Practice 1: Add

Simplify

Practice 2: Add with Missing Powers

Simplify

Practice 3: Subtract

Simplify

Practice 4: Multiply Monomials

Simplify

Practice 5: Multiply a Monomial and Polynomial

Expand

Practice 6: Multiply Two Polynomials

Expand and simplify

Practice 7: Predict Before Expanding

Without fully multiplying, determine the degree and leading coefficient of

Practice 8: Find the Error

A student writes

Identify the error and give the correct result.

Practice 9: Structural Check

A student claims

Without redoing the entire multiplication first, explain how the degree tells you immediately that the expansion must contain an error.

Practice Answers

Practice 1

Practice 2

Practice 3

Change every sign in the second polynomial:

Therefore,

Practice 4

Practice 5

Thus,

Practice 6

Practice 7

The degrees are \(6\) and \(3\), so the product has degree

The leading coefficients are \(5\) and \(-4\), so the product's leading coefficient is

Practice 8

The sign of every term in the second polynomial must change:

Therefore,

Practice 9

The factors have degrees \(3\) and \(2\), so the product must have degree \(5\).

Its leading term must be

The claimed expansion begins with \(6x^4\), so the leading multiplication was performed incorrectly.

What to Remember

Polynomial arithmetic is organized by powers and coefficients.

• Addition: add coefficients of like powers.
• Subtraction: add the additive inverse of the second polynomial, then combine like terms.
• Multiplication: multiply every term in one factor by every term in the other, then combine like terms.

For nonzero polynomials,

The leading coefficient of a product is the product of the leading coefficients, while the constant term of a product is the product of the constant terms.

Addition and subtraction work almost directly on coefficient lists. Multiplication combines shifted coefficient products, a process that can be described formally as convolution.

Where We Go Next

Polynomial multiplication works because multiplication distributes across addition. That distributive structure produces recurring patterns that can make many expansions much faster.

The next lesson, Distribution and Special Products, develops those patterns directly, including efficient methods for multiplying binomials and recognizing important algebraic identities.