Explainer: Dividing Polynomials by Monomials

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.

Monomial

A monomial is an expression that contains a single term composed of a product of constants and variables. Examples of monomials are

  • ๐‘ฅ๐‘ฆ๏Šจ,
  • 3๐‘ฅ๐‘ฆ๏Šซ,
  • โˆ’6๐‘Ž๐‘๐‘๏Šซ,
  • ๐‘˜.

Polynomial

A polynomial is an expression that contains one or more monomials. Examples of polynomials would be

  • ๐‘ฅ+5๐‘ฅ+2๏Šจ,
  • 5๐‘ฅ๐‘ฆ+๐‘ฆ๐‘ง๏Šจ,
  • 12๐‘Ž๐‘+๐‘๐‘+๐‘๐‘‘+๐‘‘+10๏Šฉ.

Note that monomials, and therefore polynomials, do not contain variables raised to negative exponents. We also need to recap the quotient rule of exponents.

Key Rule: Quotient Rule of Exponents

For the quotient of two exponential expressions, with the same base, we have that ๐‘ฅรท๐‘ฅ=๐‘ฅ๏Œบ๏Œป๏Œบ๏Šฑ๏Œป or, equivalently, ๐‘ฅ๐‘ฅ=๐‘ฅ.๏Œบ๏Œป๏Œบ๏Šฑ๏Œป

Let us now look at the process of dividing a polynomial by a monomial. Consider the expression 4๐‘ฅ๐‘ฆ+8๐‘ฅ๐‘ฆโˆ’2๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆ.๏Šจ๏Šฉ

The first thing to note is that we can convert the expression into three rational expressions that consist of a monomial divided by a monomial: 4๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆ+8๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆโˆ’2๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆ.๏Šจ๏Šฉ

We can then simplify each of the constants: 4๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆ+8๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆโˆ’2๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆ.๏Šจ๏Šจ๏Šช๏Šฉ

This leaves us with 2๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆโˆ’๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ.๏Šจ๏Šฉ

We can now simplify the variables using the quotient rule of exponents; we get 2๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆโˆ’๐‘ฅ๐‘ฆ.(๏Šจ๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง)(๏Šฉ๏Šฑ๏Šง)

This simplifies to 2๐‘ฅ๐‘ฆ+4๐‘ฅ๐‘ฆโˆ’๐‘ฅ๐‘ฆ.๏Šง๏Šฆ๏Šฆ๏Šฆ๏Šฆ๏Šจ

At this point, recall that any nonzero variable raised to the power of zero is equal to 1 (๐‘ฅ=1๏Šฆ), and any variable raised to the power of 1 is the variable itself (๐‘ฅ=๐‘ฅ๏Šง), which means our expression can be further simplified to 2๐‘ฅ+4โˆ’๐‘ฆ.๏Šจ

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 out as follows: 4๐‘ฅ2๐‘ฆ2๐‘ฅ๐‘ฆ+8๐‘ฅ๐‘ฆ2๐‘ฅ๐‘ฆโˆ’2๐‘ฅ๐‘ฆ32๐‘ฅ๐‘ฆ.๏Šจ๏Šง๏Šช๏Šจ

This looks very busy, but looking carefully, we can see that we are left with 2๐‘ฅ+4โˆ’๐‘ฆ.๏Šจ

Let us now look at some examples, solving these using a variety of different methods.

Example 1: Simplifying a Monomial Divided by a Monomial

Find the quotient of 21๐‘Ž๐‘7๐‘Ž๐‘๏Šฌ.

Answer

To find the quotient, we need to divide the dividend (the top number) by the divisor (the bottom number). If we start by simplifying the constants, we get 3๐‘Ž๐‘๐‘Ž๐‘.๏Šฌ

If we then apply the quotient rule of exponents, we get 3๐‘Ž๐‘,(๏Šฌ๏Šฑ๏Šง)(๏Šง๏Šฑ๏Šง) which simplifies to 3๐‘Ž๐‘๏Šซ๏Šฆ.

Remember that any nonzero variable raised to the power of zero is equal to 1. Therefore, the expression simplifies to 3๐‘Ž๏Šซ.

Example 2: Simplifying a Polynomial Divided by a Simple Monomial

Find the quotient of โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ๐‘ฅ๏Šญ๏Šฉ๏Šจ.

Answer

To find the quotient, we need to divide the dividend (the top number) by the divisor (the bottom number). In this question, we need to divide each of the terms in the polynomial by ๐‘ฅ. This gives us โˆ’43๐‘ฅ๐‘ฅ+12๐‘ฅ๐‘ฅโˆ’6๐‘ฅ๐‘ฅ.๏Šญ๏Šฉ๏Šจ

Here we have that ๐‘ฅรท๐‘ฅ=๐‘ฅ๏Šญ๏Šฌ, ๐‘ฅรท๐‘ฅ=๐‘ฅ๏Šฉ๏Šจ, and ๐‘ฅรท๐‘ฅ=๐‘ฅ๏Šจ, which means we can simplify the expression to โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ.๏Šฌ๏Šจ

Example 3: Simplifying a Polynomial Divided by a Monomial

Find the quotient of 26๐‘Ž๐‘โˆ’9๐‘Ž๐‘๐‘Ž๐‘๏Šญ๏Šซ๏Šฉ๏Šซ๏Šจ.

Answer

To find the quotient, we need to divide the dividend (the top number) by the divisor (the bottom number). In this question, we need to divide each of the terms in the polynomial by the monomial ๐‘Ž๐‘๏Šจ. If we start by dividing each of the terms by ๐‘Ž๏Šจ, we get 26๐‘Ž๐‘โˆ’9๐‘Ž๐‘๐‘,๏Šซ๏Šซ๏Šซ and dividing each of the terms through by ๐‘, we get 26๐‘Ž๐‘โˆ’9๐‘Ž๐‘.๏Šซ๏Šช๏Šช

When taking this approach to solving the problem, it can be helpful to show our working by canceling from the top and the bottom of the expression, that is, 26๐‘Ž7๐‘5โˆ’9๐‘Ž3๐‘5๐‘Ž๐‘.๏Šซ๏Šช๏Šง๏Šช๏Šจ

This method helps us keep track of the exponents of the variables, and we can see that the expression simplifies to 26๐‘Ž๐‘โˆ’9๐‘Ž๐‘,๏Šซ๏Šช๏Šช as before.

Before moving on to a slightly different example, let us look at an example where the monomial divisor is negative. When this is the case, we have to be particularly careful to check the signs of the resulting terms.

Example 4: Simplifying a Polynomial Divided by a Negative Monomial

Find the quotient of 23๐‘ฅ๐‘ฆ+49๐‘ฅ๐‘ฆ+41๐‘ฅ๐‘ฆโˆ’๐‘ฅ๐‘ฆ๏Šซ๏Šฉ๏Šช๏Šฉ๏Šฉ๏Šฉ.

Answer

Notice here that the monomial divisor is negative. The simplest way to approach this question is to multiply the top and the bottom of the expression through by โˆ’1 to ensure the denominator is positive. If we do this, we reverse the signs of each of the terms: โˆ’23๐‘ฅ๐‘ฆโˆ’49๐‘ฅ๐‘ฆโˆ’41๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ.๏Šซ๏Šฉ๏Šช๏Šฉ๏Šฉ๏Šฉ

We can now divide each of the terms of the polynomial by ๐‘ฅ๐‘ฆ, which gives โˆ’23๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆโˆ’49๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆโˆ’41๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ.๏Šซ๏Šฉ๏Šช๏Šฉ๏Šฉ๏Šฉ

Canceling from the top and the bottom of each rational expression gives โˆ’23๐‘ฅ5๐‘ฆ3๐‘ฅ๐‘ฆโˆ’49๐‘ฅ4๐‘ฆ3๐‘ฅ๐‘ฆโˆ’41๐‘ฅ3๐‘ฆ3๐‘ฅ๐‘ฆ.๏Šช๏Šจ๏Šฉ๏Šจ๏Šจ๏Šจ

The expression, therefore, simplifies to โˆ’23๐‘ฅ๐‘ฆโˆ’49๐‘ฅ๐‘ฆโˆ’41๐‘ฅ๐‘ฆ.๏Šช๏Šจ๏Šฉ๏Šจ๏Šจ๏Šจ

Let us finish by looking at an example of a more general problem where the division of a polynomial by a monomial arises as part of the solution.

Example 5: Finding an Expression for the Height of a Triangle

The area of a triangle is ๏€น12๐‘ฅ+4๐‘ฅ๏…๏Šจ cm2, and its base is 4๐‘ฅ m. Write an expression for its height.

Answer

Remember that the area of a triangle is equal to 12 (base ร— perpendicular height). We are told an expression for the area and base of the triangle and can, thus, form the following equation: 12๐‘ฅ+4๐‘ฅ=12ร—4๐‘ฅร—โ„Ž,๏Šจ where โ„Ž is the height. To find an expression for โ„Ž, we need to rearrange the equation to make โ„Ž the subject. We start by multiplying throughout by 2 giving us 24๐‘ฅ+8๐‘ฅ=4๐‘ฅร—โ„Ž.๏Šจ

We then need to divide throughout by 4๐‘ฅ, which gives โ„Ž=24๐‘ฅ+8๐‘ฅ4๐‘ฅ.๏Šจ

We now need to simplify the expression by dividing each of the terms of the polynomial by 4๐‘ฅ giving us โ„Ž=6๐‘ฅ+2.

Key Points

To divide a polynomial by a monomial, remember the following steps:

  1. Start by dividing each of the terms of the polynomial through by the monomial.
  2. Simplify the resulting expression by simplifying any constants and using the quotient rule of exponents.
  3. It can be helpful to show your working out by canceling from the top and the bottom of the expression, or expressions if it has been separated.

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