In this explainer, we will learn how to multiply an algebraic expression by a monomial.

We first recall that we can multiply monomials together by multiplying the coefficients and adding the exponents of the shared variables together. For example, we find the product of and by first rearranging the product as follows:

We can evaluate this product by multiplying the coefficients 5 and 3 and adding the exponents of the shared variables and . This gives

This process allows us to multiply any two monomials. In fact, we can use this same process to find the product of any number of monomials. It is worth noting that we use the properties of multiplication to rearrange this product. In particular, we use the associativity and commutativity properties to rearrange the product such that the like factors are multiplied together.

This idea of using the properties of multiplication to help simplify algebraic expressions can also be extended to the distributive property of multiplication over addition. We recall this tells us that if , , , then

In other words, multiplication distributes over addition. An example of this is the area model.

We note that the smaller rectangles have areas and and the combined rectangle has area . Since the combined rectangle has area equal to the sum of its parts, we have

Similarly, for subtraction, we note that we can model as the area of a rectangle with sides of length and .

The combined rectangle has area , and the smaller rectangles have area and . So,

We then subtract from both sides of the equation to see

The same will be true for algebraic expressions. For example, we can multiply by by distributing over the addition. This gives

This works since monomials are the product of constants and variables raised to integer exponents. Since our variables are rational, we can say that these monomials are output rational numbers. Hence, all of the properties of the rational numbers must apply to them.

Letβs see an example of distributing over subtraction.

### Example 1: Multiplying a Binomial by a Constant

Calculate .

### Answer

We first recall that we can multiply an algebraic expression by a monomial by using the distributive property of multiplication over addition. Of course, we are working with a subtraction, but we can also distribute over subtraction as follows:

In our next example, we will distribute a monomial over a binomial.

### Example 2: Multiplying a Monomial by a Binomial

Calculate .

### Answer

We first recall that we can multiply an algebraic expression by a monomial by using the distributive property of multiplication over addition. This means we need to multiply each term in the binomial by separately. This gives

We then recall that , where and are integers. Hence,

It is worth noting that the distributive property of multiplication over addition (or subtraction) works for any number of terms. For example, if , , , and , then

Extending this to algebraic expressions allows us to multiply every term in the sum by the first factor. Letβs see an example of applying this property to multiply a trinomial by a monomial.

### Example 3: Multiplying a Monomial by a Trinomial

Expand and simplify the expression .

### Answer

We first recall that we can multiply an algebraic expression by a monomial by using the distributive property of multiplication over addition. This means we need to multiply every term in the trinomial by . This gives us

We then recall that . Hence,

In our next example, we will see how we can apply the multiplication process to determine an expression for an area of a triangle whose base and height are given as expressions in .

### Example 4: Finding the Area of a Triangle by Multiplying an Algebraic Expression by a Monomial

A triangle has a height of and a base of . Find the area of the triangle in terms of .

### Answer

We first recall that the area of a triangle is equal to one-half of the length of its base multiplied by its perpendicular height. This means its area is given by the expression . We can simplify this by first noting that , giving us

We can rewrite this as a binomial by distributing over the parentheses. We have

We then recall that and . Hence,

In our next example, we will determine an expression for a region given in a diagram by using the distribution of monomials.

### Example 5: Finding the Area of a Compound Shape by Multiplying Algebraic Expressions and Monomials

Find the area of the shaded region.

### Answer

We begin by noting that the area of the shaded region will be the area of the larger rectangle, with the area of the smaller rectangle removed. We can determine each of these areas separately.

First, letβs isolate the larger rectangle.

The area of a rectangle is length times width. So, its area is . We can distribute over the parentheses to get

We then recall that , so . Hence,

Second, letβs isolate the smaller rectangle.

Multiplying its length by its width gives us an area of .

We need to subtract the area of the smaller rectangle from the area of the larger rectangle to find an expression for the area of the shaded region.

This gives us . Finally, we can combine like terms, so we get

In our next example, we will apply this process to multiply a trinomial by a monomial with multiple variables.

### Example 6: Multiplying a Trinomial by a Monomial with Multiple Variables

Expand and simplify the expression .

### Answer

We first recall that we can multiply an algebraic expression by a monomial by using the distributive property of multiplication over addition and subtraction. This means we need to multiply every term in the trinomial by . This gives us

We can then evaluate each product by recalling that we multiply the coefficients and add the exponents of the shared variables together. We get

In our final example, we will use this process of multiplying algebraic expressions by monomials to determine the missing value of a constant that equates two expressions.

### Example 7: Finding an Unknown by Comparing the Result after the Simplification of the Multiplication Problem

Find the value of the constant , given that is equivalent to .

### Answer

Since both expressions are equivalent, we can start by distributing the monomial in the first expression in order to have it in a form that can be compared with the second expression. We do this by multiplying every term by . We get

We note that is a constant, and to multiply variables, we add their exponents. We get

Since this expression needs to be equal to , we can equate the coefficients. Doing this, we get .

Letβs finish by recapping some of the important points from this explainer.

### Key Points

- We can multiply polynomials by a monomial expression by distributing the monomial. This means we multiply every term of the polynomial by the monomial.