In this explainer, we will learn how to divide polynomials by monomials.
Before looking at an example where we divide a polynomial by a monomial, we need to recap some key rules and concepts.
Definition: Monomial
A monomial is an expression that contains a single term composed of a product of constants and variables with nonnegative integer exponents. The following are examples of monomials:
Definition: Polynomial
A polynomial is an expression that contains one monomial or more. The following are examples of polynomials:
Note that monomials, and therefore polynomials, do not contain variables raised to negative exponents. For the purpose of this explainer, we also need to recap the quotient rule of exponents.
Rule: Quotient Rule
For the quotient of two exponential expressions with the same base, we have that or, equivalently,
The scope of this explainer is to cover cases of the quotient rule where .
Let us now look at the process of dividing a polynomial by a monomial. For example, consider the expression
In this case, the polynomial and monomial are both expressed in terms of a single variable, .
First, note that we can convert this expression into two rational expressions that consist of a monomial divided by a monomial:
Then, we simplify the variables using the quotient rule of exponents, which gives us
This simplifies to
Now, recall that any nonzero variable raised to the power of zero is equal to 1, so , and any variable raised to the power of 1 is the variable itself, so . Therefore, our expression simplifies to
Another way to think of this intuitively is by considering the common factors on the top and bottom of our quotients. Note that can be expressed as ; hence, returning to our original expression, we get
We can now cancel the common factor of on the top and bottom of each quotient:
Clearly, by using this method, we reach the same answer as before.
Let us now look at an example.
Example 1: Dividing a Single-Variable Polynomial by a Single-Variable Monomial
Simplify .
Answer
To find the quotient, we need to divide the dividend (the top expression) by the divisor (the bottom expression). Here, both the dividend and divisor are expressed in terms of a single variable, .
First, we convert the given expression into three rational expressions by dividing each term in the polynomial by as follows:
Next, we simplify the variables using the quotient rule, which gives us
This simplifies to
Since and , our final answer can be simplified to
We will also meet quotient expressions involving multiple variables. In the simplest cases, we have a multivariable numerator and a single-variable denominator. For example, consider the expression
The numerator is expressed in terms of two variables, and , whereas the denominator involves just . We can convert the given expression into two rational expressions by dividing each term in the polynomial by the monomial :
Then, we simplify the two terms using the quotient rule of exponents. Note that, in cases like this, the variables are undisturbed by the division operation. Therefore, we get which simplifies to
Clearly, this answer takes a very similar form to the one obtained for a single variable in the first example.
Let us now try an example of this type.
Example 2: Dividing a Multivariable Polynomial by a Single-Variable Monomial
Simplify .
Answer
To find the quotient, we need to divide the dividend (the top expression) by the divisor (the bottom expression). In this case, the dividend is expressed in terms of two variables, and , while the divisor is expressed only in terms of .
First, we convert the given expression into two rational expressions by dividing each term in the polynomial by the monomial :
We can then simplify each of the constants:
This leaves us with
Next, we simplify the variables by applying the quotient rule, giving us which further simplifies to
Since , we get , and this is our final answer.
Other quotient expressions involve multiple variables in both the numerator and denominator, including ones where the divisor monomial has a coefficient greater than 1. However, we can still follow the same method, as long as we remember to simplify the constants as well as the variables. For example, consider the more complex expression
In this case, the polynomial and monomial are both expressed in terms of two variables, and . This time, we convert the given expression into three rational expressions by dividing each term in the polynomial by the monomial :
We can then simplify each of the constants:
This leaves us with
We can now simplify the variables using the quotient rule of exponents; we get
This simplifies to
Again, recall that any nonzero variable raised to the power of zero is equal to 1 and any variable raised to the power of 1 is the variable itself, which means that our expression can be further simplified to
In reality, rather than explicitly quoting the quotient rule of exponents, it is perfectly acceptable to simplify the expression by canceling from the top and the bottom. For example, once we separated the expression into individual rational expressions, we could have shown our working as follows:
Although this might seem very busy, by looking carefully, we can see that we are left with
In cases of a multivariable denominator, it may seem easier to simplify one variable at a time; let us see an example of this now.
Example 3: Dividing a Multivariable Polynomial by a Multivariable Monomial
Find the quotient of .
Answer
To find the quotient, we need to divide the dividend (the top expression) by the divisor (the bottom expression). Here, both the dividend and divisor are expressed in terms of two variables, and . We need to divide each of the terms in the polynomial by the monomial .
If we start by dividing each of the terms by , we get which simplifies to
Then, dividing each of the terms through by , we get which simplifies to
In some questions, we are given the final simplified expression and need to work backward to find an unknown in the original polynomial or monomial. Here is an example of this type.
Example 4: Finding an Unknown by Dividing a Polynomial by a Monomial Then Comparing with the Final Simplified Expression
Find the value of the constant , given that .
Answer
To find a quotient, we divide the dividend (the top expression) by the divisor (the bottom expression). Here, we are given the simplified quotient and are shown that it arises from dividing the polynomial by the monomial . Therefore, we must work backward from the final quotient to find the value of the constant .
To begin, notice that we can rewrite the left-hand side of the original equation as two separate terms:
Simplifying the first term by dividing through by , we get
Clearly, this matches up with the first term on the right-hand side of the original equation. This implies that the second terms must also match up, so we have
Now, we use this equation to find . We begin by simplifying by dividing through by , which means the equation becomes
Observe that we now have the monomial on both sides. Therefore, we can obtain by equating the coefficients, which is the same as canceling the on both sides. Thus, we get
We sometimes meet more general problems where the division of a polynomial by a monomial arises as part of the solution. Our final example comes from a geometric context.
Example 5: Finding an Expression for the Height of a Triangle
The area of a triangle is cm2, and its base is cm. Write an expression for its height.
Answer
First, remember the formula for the area of a triangle:
In this question, we are given expressions for the area and base of a triangle. Hence, we can substitute them into the formula to form the equation where is the height. To find an expression for , we need to rearrange the equation to make the subject. We start by multiplying throughout by 2:
We then need to divide throughout by , which gives us
Notice that the right-hand side is now in the form of a quotient, where the dividend is the polynomial and the divisor is the monomial . Therefore, we need to simplify it by dividing each of the terms of the polynomial by , which gives us
As the base was given in centimetres, the height will also be in centimetres: an expression for the height of the triangle is cm.
Let us finish by recapping some key concepts from this explainer.
Key Points
- To divide a polynomial by a monomial, remember the following steps:
- Divide each of the individual terms of the polynomial by the monomial.
- Simplify any constants.
- Simplify the variable terms using the quotient rule of exponents:
- The quotient rule of exponents can be applied to problems involving single-variable and multivariable polynomials and monomials.
- In cases of a multivariable denominator, it may seem easier to simplify one variable at a time.
- It can be helpful to show your working by canceling from the top and the bottom of each expression.
