# How to multiply multivariable polynomials

## Paying attention to like terms

Multiplying multivariable polynomials (polynomials with two or more different variables) is very similar to multiplying single-variable polynomials (those that have just one variable). You’ll just need to be careful about combining like terms. In the case of a multivariable polynomial, two terms aren’t “like terms” unless each variable has the same exponent in both of them.

For example, ???3x^2y^3??? and ???-5x^2y^3??? are like terms: In both terms, the variable ???x??? has an exponent of ???2??? and the variable ???y??? has an exponent of ???3???. However, ???-9x^2y??? and ???8xy??? are **not** like terms: In ???-9x^2y??? the variable ???x??? has an exponent of ???2???, but in ???8xy??? the variable ???x??? has an exponent of ???1???.

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Here are some reminders about multiplying polynomials.

**FOIL**

FOIL is a way to help you remember to multiply each term in the first set of parentheses by each term in the second set of parentheses. FOIL stands for **Firsts**, **Outsides**, **Insides**, **Lasts**, which is the order of the four terms in the result of the multiplication; it also indicates which terms in the given binomials are multiplied to produce each term in the result.

**Chart**

The example below shows binomial multiplication (two terms by two terms), but a chart can be used when multiplying by more than two terms as well.

???ac+ad+bc+bd???

## Multiplying multivariable polynomials step-by-step

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## Multiplying a multivariable binomial by a multivariable trinomial

Let’s look at a couple of examples where we multiply multivariable polynomials.

**Example**

Simplify the expression.

???(x-2y)(2x^3-3xy-y^2)???

A chart will be useful to make sure we distribute every term in the first set of parentheses across all the terms in the second set of parentheses.

???2x^4-3x^2y-xy^2-4x^3y+6xy^2+2y^3???

Next, we’ll rearrange the terms in descending order of powers of ???x??? (a common practice when dealing with multivariable polynomials).

???2x^4-4x^3y-3x^2y-xy^2+6xy^2+2y^3???

Now we’ll group like terms together, and then combine like terms.

???2x^4-4x^3y-3x^2y+(-xy^2+6xy^2)+2y^3???

???2x^4-4x^3y-3x^2y+5xy^2+2y^3???

Find individual terms by multiplying the polynomials, then group together like terms, then combine like terms to simplify

Let’s try another example of multiplying multivariable polynomials.

**Example**

Simplify the expression.

???(2x+3y)(x-y)+(x+y)(4x-2y)???

Multiply the first pair of binomials by using either a chart or FOIL.

???2x^2-2xy+3xy-3y^2+(x+y)(4x-2y)???

Combine like terms.

???2x^2+xy-3y^2+(x+y)(4x-2y)???

Multiply the other pair of binomials by using either a chart or FOIL.

???2x^2+xy-3y^2+4x^2-2xy+4xy-2y^2???

Group like terms together, and then combine like terms.

???(2x^2+4x^2)+(xy-2xy+4xy)+(-3y^2-2y^2)???

???6x^2+3xy-5y^2???