Lesson Explainer: Dividing Polynomials by Monomials | Nagwa Lesson Explainer: Dividing Polynomials by Monomials | Nagwa

Lesson Explainer: Dividing Polynomials by Monomials Mathematics • First Year of Preparatory School

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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:

  • π‘₯π‘¦οŠ¨
  • 3π‘₯π‘¦οŠ«
  • βˆ’6π‘Žπ‘π‘οŠ«
  • π‘˜

Definition: Polynomial

A polynomial is an expression that contains one monomial or more. The following are examples of polynomials:

  • π‘₯+5π‘₯+2
  • 5π‘₯𝑦+π‘¦π‘§οŠ¨
  • 12π‘Žπ‘+𝑏𝑐+𝑐𝑑+𝑑+10

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 π‘₯Γ·π‘₯=π‘₯ or, equivalently, π‘₯π‘₯=π‘₯.

The scope of this explainer is to cover cases of the quotient rule where π‘Žβ‰₯𝑏.

Let us now look at the process of dividing a polynomial by a monomial. For example, consider the expression 3π‘₯+π‘₯π‘₯.

In this case, the polynomial and monomial are both expressed in terms of a single variable, π‘₯.

First, note that we can convert this expression into two rational expressions that consist of a monomial divided by a monomial: 3π‘₯π‘₯+π‘₯π‘₯.

Then, we simplify the variables using the quotient rule of exponents, which gives us 3π‘₯+π‘₯.()()

This simplifies to 3π‘₯+π‘₯.

Now, recall that any nonzero variable raised to the power of zero is equal to 1, so π‘₯=1, and any variable raised to the power of 1 is the variable itself, so π‘₯=π‘₯. Therefore, our expression simplifies to 3π‘₯+1.

Another way to think of this intuitively is by considering the common factors on the top and bottom of our quotients. Note that 3π‘₯ can be expressed as 3Γ—π‘₯Γ—π‘₯; hence, returning to our original expression, we get 3Γ—π‘₯Γ—π‘₯π‘₯+π‘₯π‘₯.

We can now cancel the common factor of π‘₯ on the top and bottom of each quotient: 3Γ—π‘₯Γ—π‘₯π‘₯+π‘₯π‘₯.

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 π‘₯+7π‘₯βˆ’9π‘₯π‘₯οŠͺ.

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, π‘₯.

First, we convert the given expression into three rational expressions by dividing each term in the polynomial by π‘₯ as follows: π‘₯π‘₯+7π‘₯π‘₯βˆ’9π‘₯π‘₯.οŠͺ

Next, we simplify the variables using the quotient rule, which gives us π‘₯+7π‘₯βˆ’9π‘₯.(οŠͺ)()()

This simplifies to π‘₯+7π‘₯βˆ’9π‘₯.

Since π‘₯=1 and π‘₯=π‘₯, our final answer can be simplified to π‘₯+7π‘₯βˆ’9.

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 3π‘₯𝑦+π‘₯π‘₯.

The numerator is expressed in terms of two variables, π‘₯ and 𝑦, whereas the denominator involves just π‘₯. We can convert the given expression into two rational expressions by dividing each term in the polynomial by the monomial π‘₯: 3π‘₯𝑦π‘₯+π‘₯π‘₯.

Then, we simplify the two terms using the quotient rule of exponents. Note that, in cases like this, the 𝑦 variables are undisturbed by the division operation. Therefore, we get 3π‘₯𝑦+π‘₯,()() which simplifies to 3π‘₯𝑦+1.

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 4π‘₯π‘¦βˆ’6π‘₯𝑦2π‘₯.

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, π‘₯ and 𝑦, while the divisor is expressed only in terms of π‘₯.

First, we convert the given expression into two rational expressions by dividing each term in the polynomial by the monomial 2π‘₯: 4π‘₯𝑦2π‘₯βˆ’6π‘₯𝑦2π‘₯.

We can then simplify each of the constants: 4π‘₯𝑦2π‘₯βˆ’6π‘₯𝑦2π‘₯.

This leaves us with 2π‘₯𝑦π‘₯βˆ’3π‘₯𝑦π‘₯.

Next, we simplify the variables by applying the quotient rule, giving us 2π‘₯π‘¦βˆ’3π‘₯𝑦,()() which further simplifies to 2π‘₯π‘¦βˆ’3π‘₯𝑦.

Since π‘₯=1, we get 2π‘₯π‘¦βˆ’3π‘¦οŠ¨οŠ©οŠ­, 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 4π‘₯𝑦+8π‘₯π‘¦βˆ’2π‘₯𝑦2π‘₯𝑦.

In this case, the polynomial and monomial are both expressed in terms of two variables, π‘₯ and 𝑦. This time, we convert the given expression into three rational expressions by dividing each term in the polynomial by the monomial 2π‘₯𝑦: 4π‘₯𝑦2π‘₯𝑦+8π‘₯𝑦2π‘₯π‘¦βˆ’2π‘₯𝑦2π‘₯𝑦.

We can then simplify each of the constants: 4π‘₯𝑦2π‘₯𝑦+8π‘₯𝑦2π‘₯π‘¦βˆ’2π‘₯𝑦2π‘₯𝑦.οŠͺ

This leaves us with 2π‘₯𝑦π‘₯𝑦+4π‘₯𝑦π‘₯π‘¦βˆ’π‘₯𝑦π‘₯𝑦.

We can now simplify the variables using the quotient rule of exponents; we get 2π‘₯𝑦+4π‘₯π‘¦βˆ’π‘₯𝑦.()()()()()()

This simplifies to 2π‘₯𝑦+4π‘₯π‘¦βˆ’π‘₯𝑦.

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 2π‘₯+4βˆ’π‘¦.

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: 4π‘₯2𝑦2π‘₯𝑦+8π‘₯𝑦2π‘₯π‘¦βˆ’2π‘₯𝑦32π‘₯𝑦.οŠͺ

Although this might seem very busy, by looking carefully, we can see that we are left with 2π‘₯+4βˆ’π‘¦.

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 26π‘Žπ‘βˆ’9π‘Žπ‘π‘Žπ‘οŠ­οŠ«οŠ©οŠ«οŠ¨.

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, π‘Ž and 𝑏. We need to divide each of the terms in the polynomial by the monomial π‘Žπ‘οŠ¨.

If we start by dividing each of the terms by π‘ŽοŠ¨, we get 26π‘Ž7π‘βˆ’9π‘Ž3π‘π‘Žπ‘, which simplifies to 26π‘Žπ‘βˆ’9π‘Žπ‘π‘.

Then, dividing each of the terms through by 𝑏, we get 26π‘Žπ‘5βˆ’9π‘Žπ‘5𝑏,οŠͺοŠͺ which simplifies to 26π‘Žπ‘βˆ’9π‘Žπ‘.οŠͺοŠͺ

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 π‘˜, given that 4π‘₯𝑦+π‘˜π‘₯𝑦2π‘₯𝑦=2π‘₯𝑦+3π‘₯π‘¦οŠ«οŠ¨οŠ­οŠ©οŠ¨οŠ©οŠ«οŠ¨.

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 2π‘₯𝑦+3π‘₯π‘¦οŠ©οŠ«οŠ¨ and are shown that it arises from dividing the polynomial 4π‘₯𝑦+π‘˜π‘₯π‘¦οŠ«οŠ¨οŠ­οŠ© by the monomial 2π‘₯π‘¦οŠ¨. Therefore, we must work backward from the final quotient to find the value of the constant π‘˜.

To begin, notice that we can rewrite the left-hand side of the original equation as two separate terms: 4π‘₯𝑦2π‘₯𝑦+π‘˜π‘₯𝑦2π‘₯𝑦.

Simplifying the first term by dividing through by 2π‘₯π‘¦οŠ¨, we get 4π‘₯𝑦2π‘₯𝑦=2π‘₯𝑦.

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 π‘˜π‘₯𝑦2π‘₯𝑦=3π‘₯𝑦.

Now, we use this equation to find π‘˜. We begin by simplifying π‘˜π‘₯𝑦2π‘₯π‘¦οŠ­οŠ©οŠ¨ by dividing through by π‘₯π‘¦οŠ¨, which means the equation becomes π‘˜2π‘₯𝑦=3π‘₯𝑦.

Observe that we now have the monomial π‘₯π‘¦οŠ«οŠ¨ on both sides. Therefore, we can obtain π‘˜ by equating the coefficients, which is the same as canceling the π‘₯π‘¦οŠ«οŠ¨ on both sides. Thus, we get π‘˜2=3,π‘˜=6.

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 ο€Ή12π‘₯+4π‘₯ο…οŠ¨ cm2, and its base is 4π‘₯ cm. Write an expression for its height.

Answer

First, remember the formula for the area of a triangle: areaoftrianglebaseperpendicularheight=12Γ—Γ—.

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 12π‘₯+4π‘₯=12Γ—4π‘₯Γ—β„Ž, where β„Ž is the height. To find an expression for β„Ž, we need to rearrange the equation to make β„Ž the subject. We start by multiplying throughout by 2: 24π‘₯+8π‘₯=4π‘₯Γ—β„Ž.

We then need to divide throughout by 4π‘₯, which gives us β„Ž=24π‘₯+8π‘₯4π‘₯.

Notice that the right-hand side is now in the form of a quotient, where the dividend is the polynomial 24π‘₯+8π‘₯ and the divisor is the monomial 4π‘₯. Therefore, we need to simplify it by dividing each of the terms of the polynomial by 4π‘₯, which gives us β„Ž=24π‘₯4π‘₯+8π‘₯4π‘₯=6π‘₯+2π‘₯=6π‘₯+2π‘₯=6π‘₯+2.()()

As the base was given in centimetres, the height will also be in centimetres: an expression for the height of the triangle is 6π‘₯+2 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:
    1. Divide each of the individual terms of the polynomial by the monomial.
    2. Simplify any constants.
    3. Simplify the variable terms using the quotient rule of exponents: π‘₯Γ·π‘₯=π‘₯π‘₯=π‘₯.
  • 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.

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