Multiplying Polynomials

Quick Reference

In this section we will learn how to multiply two polynomials.

Method	Best used for	Core idea
Distributive property	Multiplication of any two polynomials	Multiply each term of the first by each term of the second
FOIL	Multiplication of two binomials only	First, Outer, Inner, Last
Long (vertical) method	Multiplication of three or more terms	Align partial products by degree in columns

The Rule

To multiply two polynomials, apply the distributive property by multiplying each term of the first polynomial by each term of the second. Then use the exponent rule

and collect any like terms.

Product of Two Monomials

To multiply two monomials, multiply the coefficients and add the exponents:

Product of a Monomial and a Polynomial

Use the distributive property to multiply the monomial by each term of the polynomial:

Product of Two Polynomials

Distribute each term of the first polynomial across all terms of the second, then collect like terms. For example, to expand , distribute and then :

Adding the results:

Example 1. Multiply .

Solution

Degree of the Product

If is a polynomial of degree and is a polynomial of degree , then the product is a polynomial of degree :

In Example 1, has degree and has degree ; the product has degree , as expected.

Long Multiplication

When multiplying polynomials with three or more terms, a structured layout analogous to long multiplication of integers is often cleaner and less error-prone. The procedure is:

Arrange both polynomials in descending order of degree. Write the one with more terms on top (the multiplicand).
Form partial products: multiply the multiplicand by each term of the multiplier separately, writing each result on its own row.
Align by degree: place each partial product so that terms of the same degree sit in the same column.
Add column by column to obtain the final result.

Key tip: if the multiplicand is missing a term of some degree, that column is simply left blank (or filled with ) to keep the alignment correct.

Example 2. Multiply by .

Solution

Arrange in descending order and form the three partial products:

Add the partial products, aligning by degree:

Therefore,

Example 3. Multiply by using the long multiplication method.

Solution

The multiplier

has no

or constant term, so those columns are empty:

Adding:

Therefore,

, confirming Example 1.

FOIL Method for Multiplying Binomials

When multiplying two binomials, a helpful mnemonic called FOIL organizes the four products that arise from the distributive property. FOIL stands for First, Outer, Inner, Last.

Consider two binomials and where and represent any algebraic terms. For example, in (3x-2)(-2x^2+5x), we have , , , and .The four FOIL products are:

F (First): Multiply the first terms of each binomial: .
O (Outer): Multiply the outer terms: .
I (Inner): Multiply the inner terms: .
L (Last): Multiply the last terms: .

Adding all four products: .

Diagram of the FOIL method for multiplying (a+b)(c+d): four arcs connect the First terms (a and c), the Outer terms (a and d), the Inner terms (b and c), and the Last terms (b and d).

Example 4. Multiply using the FOIL method.

Solution

F (First):
O (Outer):
I (Inner):
L (Last):

Adding:

, and combining like terms:

Example 5. Multiply using the FOIL method.

Solution

F (First):
O (Outer):
I (Inner):
L (Last):

Adding:

, and combining like terms:

FOIL is a useful mnemonic for binomials, but it is simply the distributive property in disguise. The core rule is always to multiply every term in the first polynomial by every term in the second. For polynomials with more than two terms, FOIL does not directly apply; use the distributive property or the long multiplication method instead.

Determining Coefficients in the Product

Consider the product of a degree-3 polynomial by a degree-2 polynomial:

The product has degree , and a clear pattern governs each coefficient:

The coefficient of in the product is the sum of all products where , with ranging over the degrees present in the first polynomial and over those in the second.

For example, to get the coefficient of , collect all pairs with : they are , , and , giving .

This rule applies to the product of any two polynomials

and provides an efficient way to find any single coefficient without expanding the entire product.

Example 6. Find the coefficient of in the product

Solution

The degree of the product is

. For the term

, we need

with

and

. The largest possible

(giving

) and the smallest is

(giving

). So

ranges from

, and the coefficient of

For comparison, the leading coefficient of the product (coefficient of

) has only one contributing pair,

and

, so it equals

Example 7. Find the coefficient of in the product

Solution

We need all pairs

with

, where

and

The coefficient of

Frequently Asked Questions

How do you multiply two polynomials?

Multiply each term of the first polynomial by every term of the second using the distributive property, then collect like terms. For two binomials, FOIL is a convenient shortcut. For longer polynomials, the long (vertical) method keeps like-degree terms aligned and reduces errors.

What is the FOIL method and when does it apply?

FOIL stands for First, Outer, Inner, Last. It is a mnemonic for the four products that arise when multiplying two binomials:

. FOIL applies only to binomials (two-term polynomials). For trinomials or longer polynomials, the full distributive property or the long multiplication method must be used.

What is the degree of the product of two polynomials?

The degree of the product equals the sum of the degrees of the factors. If

has degree

and

has degree

, then

has degree

. For example, a degree-3 polynomial times a degree-4 polynomial gives a degree-7 polynomial.

How do you find one specific coefficient in a product without expanding fully?

The coefficient of

in the product

is the sum of all products

where

. List all valid index pairs

and sum their contributions. This avoids computing every term of the product.

Does the order of multiplication matter for polynomials?

No. Polynomial multiplication is commutative:

. The result is the same regardless of which polynomial is written first. In the long multiplication layout, placing the polynomial with more terms on top usually makes the work neater, but the answer is unaffected.

How do you multiply three polynomials?

Multiply any two of them first to get an intermediate result, then multiply that result by the third. For example, to compute

, first find

, then compute

Quick Reference

The Rule

Product of Two Monomials

Product of a Monomial and a Polynomial

Product of Two Polynomials

Degree of the Product

Long Multiplication

FOIL Method for Multiplying Binomials

Determining Coefficients in the Product

Frequently Asked Questions

Quick Links

Social Media