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

Product of Two Monomials

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

$$(-4y^3)(5y^6)=(-4)(5)y^{3+6}=-20y^9.$$

Product of a Monomial and a Polynomial

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

$$3x^2(2x^3-4x+5)=6x^5-12x^3+15x^2.$$

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 $(3x-2)(2x^2-3x+4)$, distribute $3x$ and then $-2$:

\begin{aligned} 3x(2x^2-3x+4) &= 6x^3-9x^2+12x \\ -2(2x^2-3x+4) &= -4x^2+6x-8 \end{aligned}

Adding the results:

$$6x^3+(-9-4)x^2+(12+6)x-8=6x^3-13x^2+18x-8.$$

Degree of the Product

In Example 1, $4x^2-3x+5$ has degree $2$ and $2x^3-x$ has degree $3$; the product has degree $2+3=5$, 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:

1. Arrange both polynomials in descending order of degree. Write the one with more terms on top (the multiplicand).
2. Form partial products: multiply the multiplicand by each term of the multiplier separately, writing each result on its own row.
3. Align by degree: place each partial product so that terms of the same degree sit in the same column.
4. 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 $0$) to keep the alignment correct.

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 $(a+b)$ and $(c+d)$ where $a,b,c$ and $d$ represent any algebraic terms. For example, in (3x-2)(-2x^2+5x), we have $a=3x$, $b=-2$, $c=-2x^2$, and $d=5x$.The four FOIL products are:

1. F (First): Multiply the first terms of each binomial: $a\cdot c=ac$.
2. O (Outer): Multiply the outer terms: $a\cdot d=ad$.
3. I (Inner): Multiply the inner terms: $b\cdot c=bc$.
4. L (Last): Multiply the last terms: $b\cdot d=bd$.

Adding all four products: $(a+b)(c+d)=ac+ad+bc+bd$.

Determining Coefficients in the Product

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

\begin{aligned} &(a_3x^3+a_2x^2+a_1x+a_0)(b_2x^2+b_1x+b_0) \\ &= a_3b_2\,x^5+(a_3b_1+a_2b_2)x^4+(a_3b_0+a_2b_1+a_1b_2)x^3 \\ &\quad+(a_2b_0+a_1b_1+a_0b_2)x^2+(a_1b_0+a_0b_1)x+a_0b_0. \end{aligned}

The product has degree $3+2=5$, and a clear pattern governs each coefficient:

For example, to get the coefficient of $x^3$, collect all pairs $(j,k)$ with $j+k=3$: they are $(3,0)$, $(2,1)$, and $(1,2)$, giving $a_3{b}_0+a_2{b}_1+a_1{b}_2$.

This rule applies to the product of any two polynomials

$$(a_m{x}^m+\cdots +a_0)(b_n{x}^n+\cdots +b_0)$$

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

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: $(a+b)(c+d)=ac+ad+bc+bd$. 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 $A$ has degree $n$ and $B$ has degree $m$, then $A\cdot B$ has degree $n+m$. 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 $x^i$ in the product $(a_m{x}^m+\cdots +a_0)(b_n{x}^n+\cdots +b_0)$ is the sum of all products $a_j{b}_k$ where $j+k=i$. List all valid index pairs $(j,k)$ 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: $A\cdot B=B\cdot A$. 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 $A\cdot B\cdot C$, first find $D=A\cdot B$, then compute $D\cdot C$.