Dividing Polynomials: Definition, Examples.

One of the most challenging yet interesting things to do would be to know how to divide polynomials. This might seem like an intimidating operation to perform. However, it would be surprisingly simple if one could recall all the basic formulae and rules regarding the long division of integers. 

At first, one needs to understand what a monomial is. Then, they should know about how it is closely related to polynomials. Finally, you should have an idea of what happens between two polynomials in an equation. 

Polynomials

A polynomial is identified as an expression used in algebra (an important branch of mathematics). It is constructed upon two or more terms that are added, multiplied, or subtracted. It contains constants, exponents, variables, and coefficients. It also has operators namely subtraction and addition.

Polynomials must not contain fractional or negative exponents. Some of the examples are given below:

• 8 x² – 5 x + 8
• 7 y³ + 12 y² – 2 y + 1
• 6 z² – z – 1
• 2 x³ + 6 x² – 4 x + 10

Polynomials are of Three Major Types. They are Monomial, Binomial and Trinomial.

1. Monomial

Monomials are described as algebraic expressions that contain only one term. Some examples of monomials are, 

5 x³, 7 x, – 2 y, 2z, etc. 

2. Binomials

Binomials are mathematical expressions that consist of two terms in it. It is either separated by a subtraction (-) or an addition symbol (+).

Few examples are, 5 x² +6, y³ – 2y, 2 x y + 4 y, x³ – 5, etc.

3. Trinomials

It is nothing but an algebraic expression that contains three terms in it. For example,

9 x³ – 2 x² +  7, 4 y² –  2y – 2, 12 x – 6 y + 2, 3 x y + 2 z³ – 19.

How to Divide Polynomials?

In arithmetic operations, division is utilized to split a quantity into equal amounts. Division is commonly considered reverse multiplication or repeated subtraction.

In math, there are two ways of dividing polynomials. They are synthetic division and dividing polynomials long division method. The synthetic method is known to be a fun way of dividing polynomials. On the contrary, the long division method is tiresome and consists of a long process. 

Types of Polynomial Division

After having a brief look at how to divide polynomials, the next thing one should know is what the different types of polynomial divisions available are. There are 4 basic types of division in polynomials. They are:

• Dividing a monomial using another monomial
• Dividing polynomials by monomials
• Dividing polynomials by binomials
• Dividing a polynomial using another polynomial

Dividing a Monomial by Another Monomial

It is common to divide the coefficient at first. Later on, one needs to apply the quotient law to the variables while processing the division of a monomial using another monomial.

The quotient law is x^m/xⁿ = x^m-n. 

An important note is that any variable or number that contains a power of 0 is 1. To give an example, y0 = 1. Let us have a wider look at more examples:

1. Divide 20 y² by 10 y

Solution: 1^st we divide the coefficients of the given terms.

20/10 = 2

Then divide the variables with the help of the quotient rule:

y²/y = y^2-1

= y

The next step is, multiplying the quotient of the variables by the quotient of the coefficients:

= 2 y

In an alternative way, 20 y²/10 y =5 * 2 * 2 * y * y/5 * 2 * y

Canceling y, 5, and 2 since they are common terms and are present in both the numerator as well as the denominator,

20 y²/10 y = 2 y

2. Divide 25 x³ y³ z by 5 x y² z

Solution: Start by dividing the coefficients normally and then use the quotient law to divide the variables,

25 x³ y³ z / 5 x y² z = (25/5)x^3-1 y^3-2 z^1-1

= 5 x² y¹ z⁰

= 5 x² y (final answer)

3. Divide 63 x² y³ z² by 7 x y² z

Solution: Firstly, start dividing the coefficient normally. After doing so, to divide the variables, utilize the quotient law,

63 x² y³ z² / 7 x y² z = (63/7) x^{2 -1} y^3-2 z^{2 -1}

= 9 x y z (final answer)

Dividing Polynomials by Monomials

It is simple to divide polynomials using a monomial. First, one needs to divide each term of the polynomial separately with a monomial. After that, to obtain the answer, we simply add the quotient of each operation.

Let us understand thoroughly using a few examples:

1. Divide 12 y² – 6 x y + 9x² y z by 3 y

(12 y² – 6 x y + 9 x² y z)/3 y = 12 y²/3 y – 6 x y/3 y + 9 x² y z/3 y

Answer = 4 y – 2 x  + 3 x² z

2. Divide 40 x y² z³ + 25 x³ y² z – 15 x² y z² by 5 x y z

(40 x y² z³ + 25 x³ y² z – 15 x² y z² )/5 x y z

= 40 x y² z³/5 x y z + 25 x³ y² z/5 x y z – 15 x² y z²/5 x y z

Answer: 8 y z² + 5 x² y – 3 x z

3. Divide 32 x y z³ + 48 x³ y³ z – 16 x² y z² by 8 x z

(32 x y z³ + 48 x³ y³ z – 16 x² y z² )/8 x z

= 32 x y z³/8 x z + 48 x³ y³ z/8 x z – 16 x² y z²/8 x z

Answer: 4 y z² + 6 x² y³ – 2 x y z

4.  Divide 8 m⁴ n² + 16 m n³ – 12 m³ n² by 4 m n²

8 m⁴ n² + 16 m n³ – 12 m³ n²/ 4 m n² = 8 m⁴ n²/4 m n² + 16 m n³/4 m n² – 12 m³ n²/4 m n²

Answer: 2 m³ + 4 n – 3 m²

5. Divide b³ + a b² – a³ b² by b²

(b³ + a b² – a³ b²)/b²

= b³/b² + a b²/b² – a³ b²/b²

Answer: b + a – a³

Polynomial Division Algorithm

Consider m (y) and n (y) are any two polynomials with n y≠0. The polynomials q (y) and r (y) can be found by:

m (y) = n (y) * q (y)+ r (y) 

Here, 

r (y) = 0 or degree of r (y) < degree of n (y)

The result obtained is known as the division algorithm for polynomials.

As a result of the previous example, one can verify this algorithm as:

m (y)=3 y³ + y² + 2 y + 5 

n (y) = y² + 2 y + 1 

Quotient = q (y) = 3 y – 1

Remainder = r (y) = 9 y + 10

Now, 

n (y) * q (y) + r (y) = (y² + 2 y + 1) * (2 y – 5) + (9 y + 10) 

= 3 y³ + 6 y² + 3 y – 5 y² – 10 y – 5 + 9 + 10 

= 3 y³ + y² + 2 y + 5 

=  m (y) 

Therefore, the division algorithm has been verified.

Dividing Polynomials Using Long Division

Dividing polynomials using long division is a very complex technique. However, it is a very reliable and suitable method to obtain results while dividing polynomials. This method is practical for all polynomial equations. 

This process is very similar to dividing numbers or integers. It requires the assistance of the long-division method. The following procedures should be followed while dividing polynomials using long division:

1. Firstly, rearrange the divisor in descending order. Then arrange the divider in the same way.
2. Secondly, divide the first term of the dividend by the divisor’s first term. This will help obtain the first quotient term.
3. Now identify the product of the entire divisor’s term. Then find the first term quotient. After doing so, subtract the answer of the dividend.
4. If it still contains a remainder, keep on proceeding to the 3^rd step until obtaining zero as the remainder. If not, we will end up obtaining a solution having a lesser degree than that of the divisor.

Let us see an example of the same:

Example: Using the long division technique, divide the given polynomials: 3 y³-8 y+5 by y-1

Dividing Polynomials With Remainders

To understand the concept of dividing polynomials with remainders, let us consider:

a (y) = 5 y² – 6 y + 10 y – 2 and b (y) = 15 y³ + 2 x 

In this, while dividing a by b, one can determine the remainder polynomial (r) and unique quotient polynomial (q) which is capable of satisfying the following equation:

a (y)/b (y) = q (y) + r (y)/b (y) 

Here the degree of b (y) is higher than the degree of r (y).

Conclusion

From this article, one can have a detailed understanding and idea of what exactly is a polynomial. It also provides knowledge on how to divide polynomials. We then discussed dividing polynomials by monomials, dividing polynomials by long division, etc. Finally, we have seen the topic named polynomials with remainders. Other than that, examples are also provided for all the discussed concepts. This would help students understand all these concepts in a much clearer way.

Frequently Asked Questions?

1. How do you multiply and divide polynomials?

Multiplying and dividing polynomials is a little bit different than multiplying and dividing integers.

The first step is to remember that you’re multiplying or dividing two polynomials. That means that the first thing you need to do is “combine” them into one single polynomial so they can be multiplied or divided together.

To do this, you just add the coefficients of the variables in both polynomials together. So if you have a term with an x^2 coefficient in one and a 5x coefficient in another, what you’ll end up with is a term with an x^5 coefficient.

That’s all there is to it! Now you can multiply or divide that new polynomial by anything else, just like any other polynomial would be.

2. What is the purpose of the division of polynomials?

The purpose of the division of polynomials is to find the quotient of two polynomials.

To divide two polynomials, we could just multiply them and divide them by the highest power in each term. However, there are some cases where we can’t do this because one of the factors is zero. We need a way to make sure that these cases don’t affect our result, so we use the division algorithm for polynomials.

3. Can you always use synthetic division for dividing polynomials?

 Yes, you can always use synthetic division for dividing polynomials.

The reason is that synthetic division is a method of finding the quotient of two polynomials with rational coefficients. So if both of your polynomials have rational coefficients, then it will work.

4. How to divide polynomials using synthetic division?

Synthetic division is a way to divide two polynomials. To do so, you will need to write them in the form ax + b. Then, all you have to do is find x such that ax + b = 0.

We’ll go through an example:

Let’s say we want to divide 10x^2 + 5x – 3 by 2x^3 – 2. We would first write out the polynomials in the form ax + b and then multiply them together:10x^2 + 5x – 3 * 2x^3 – 2 = (10 * 2)x^2 * 3 * (-2) = 20x^2 – 6x + 6 (Notice how we multiplied each term by 10 before we multiplied them together.)

Now, we just need to find x such that 20x^2 – 6x + 6 = 0. The answer is 4/5. So now we know that 10(4/5) = 40/5, so 10(40/5) = 400/5, so 400/5 = 80. That means that our final answer is 80/5 or 16.

5. How to solve dividing polynomials?

There are two main ways to solve dividing polynomials:

-If the divisor is a monomial (a single term), then you can divide both the numerator and denominator by that monomial.

-If the divisor is not a monomial, but has terms of all the same degree, then you can multiply both sides by the highest power of x that appears in either term of that polynomial.