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• Long Division Polynomial – Method, Definition With Examples

Long Division Polynomial – Method, Definition With Examples

Jo-ann Caballes

Updated on January 7, 2024

Welcome to another exciting journey through the world of mathematics brought to you by Brighterly, your trusted partner in making math learning enjoyable and accessible for children. Today, we delve deep into an intriguing topic, the Polynomial Long Division. Our aim is to demystify this seemingly complex concept and make it as easy and understandable as possible. In this insightful article, we will elucidate the method, provide a comprehensive definition, and walk you through a number of practical examples. So, put on your thinking cap and join us on this fascinating mathematical adventure!

What is Polynomial Long Division?

Polynomial Long Division is an intricate yet exciting mathematical procedure used to divide a polynomial by another polynomial of the same or lower degree. This method is similar to the long division you learned with numbers, but here we apply it to polynomials. It’s a significant skill that enhances logical reasoning and builds a strong foundation for future mathematical endeavors.

Definition of Polynomial Long Division

In simple terms, Polynomial Long Division is the process of dividing a given polynomial, known as the dividend, by another polynomial, called the divisor. The result is a quotient, and sometimes, a remainder is left over if the division isn’t exact.

Definition of Dividend, Divisor, Quotient, and Remainder in the Context of Polynomial Division

In the context of polynomial division:

• Dividend: The polynomial that is to be divided.
• Divisor: The polynomial by which the dividend is divided.
• Quotient: The result obtained from the division of the dividend by the divisor.
• Remainder: What’s left over from the dividend after division by the divisor.

Procedure for Polynomial Long Division

The procedure for Polynomial Long Division is systematic. You start by arranging the dividend and divisor in decreasing powers. Then, divide the leading term (highest power) of the dividend by the leading term of the divisor. The result is the first term of the quotient. Multiply the divisor by the first term of the quotient, subtract the product from the dividend to obtain a new dividend and repeat the process until you can’t divide any more.

Properties of Polynomial Long Division

Some unique properties of Polynomial Long Division include:

1. If a polynomial is divided by its factor, the quotient is another factor, and there is no remainder.
2. The degree of the remainder is always less than the degree of the divisor.
3. The degree of the divisor is less than or equal to the degree of the dividend.

Key Steps in Polynomial Long Division

Polynomial Long Division can be summarized in five key steps:

1. Arrange the dividend and divisor in descending order of powers.
2. Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
3. Multiply the divisor by the first term of the quotient.
4. Subtract the product from the dividend to get a new dividend.
5. Repeat steps 2 to 4 until the degree of the dividend is less than the divisor.

Benefits and Limitations of Polynomial Long Division

Polynomial Long Division is a powerful mathematical tool with several benefits:

1. It strengthens conceptual understanding.
2. It helps in simplifying complex polynomial expressions.
3. It provides a foundation for understanding advanced mathematical concepts.

But like any other concept, it has its limitations too:

1. It can be time-consuming with higher degree polynomials.
2. It can be complex for beginners.

Differences Between Polynomial Long Division and Numerical Long Division

While the principles of Polynomial Long Division and Numerical Long Division are similar, there are some differences. In Numerical Long Division, we divide numbers, whereas in Polynomial Long Division, we divide polynomials. Also, Polynomial Long Division may have remainders that are polynomials rather than single numbers.

Writing Equations for Polynomial Long Division

In writing equations for Polynomial Long Division, remember that the product of the divisor and the quotient, added to the remainder, equals the dividend. This is similar to the relationship in numerical division.

Examples of Polynomial Long Division

Understanding examples can be the best way to learn Polynomial Long Division. For instance, if you divide x^3 – 3x^2 + x + 2 by x – 1, you’ll find that the quotient is x^2 – 2x + 3 and the remainder is 5.

Practice Problems on Polynomial Long Division

Here at Brighterly, we firmly believe that practice is key to mastering any mathematical concept, and Polynomial Long Division is no exception. To facilitate your learning process, we’ve curated a few practice problems along with their solutions to help you get started. Remember, the goal here is not just to find the right answer, but to understand the steps and logic behind each solution.

1. Problem: Divide the polynomial 4x^3 – 6x^2 + 7x – 3 by the polynomial x – 2.

   Solution: Arranging the terms in decreasing order of their degree and following the steps of Polynomial Long Division, we get:

   Quotient = 4x^2 + 2x + 11, and Remainder = 19.

2. Problem: Divide the polynomial 3x^4 – 5x^3 + 2x^2 – 7x + 1 by the polynomial x^2 + 2.

   Solution: Following the Polynomial Long Division process:

   Quotient = 3x^2 – 11x + 24, and Remainder = -47x + 49.

3. Problem: Divide the polynomial 2x^5 – x^3 + 6x^2 – 4 by the polynomial x^2 – 1.

   Solution: Performing the Polynomial Long Division:

   Quotient = 2x^3 + 2x^2 – 4x + 4, and Remainder = 4.

Conclusion

We’ve embarked on a thrilling mathematical journey through the concept of Polynomial Long Division, haven’t we? With Brighterly, you’ve traversed from understanding the method to grappling with the definitions, and from there, diving into examples, and finally practicing the problems yourself. Just remember, like every new concept, mastering Polynomial Long Division requires practice. As daunting as it may seem at first, the rewards it brings in developing a solid mathematical foundation are truly invaluable.

So, don’t let initial struggles deter you. Keep practicing, keep questioning, and you’ll surely master Polynomial Long Division soon. Brighterly is here with you, every step of the way, illuminating your path to mathematical proficiency. Remember, at Brighterly, we believe mathematics isn’t just about numbers, equations, or algorithms: it’s about understanding. It’s about experiencing that unique joy when a complex concept finally ‘clicks’ and makes perfect sense. Let’s continue this exciting journey of exploration and discovery together!

Frequently Asked Questions on Polynomial Long Division

What are the prerequisites for learning Polynomial Long Division?

Before starting with Polynomial Long Division, it is essential to have a good understanding of basic arithmetic operations (addition, subtraction, multiplication, division), the concept of variables, degrees of a polynomial, and basic polynomial operations.

Why do we use Polynomial Long Division?

Polynomial Long Division is used to simplify complex polynomial expressions or to break them down into simpler, more manageable parts. It plays a key role in various branches of mathematics and engineering and is essential for higher-level mathematics.

Can Polynomial Long Division be used with any type of polynomial?

Yes, Polynomial Long Division can be used to divide any type of polynomial (be it linear, quadratic, cubic, and so on) by another polynomial of the same or lower degree.

What if I make a mistake while performing Polynomial Long Division?

Mistakes are part of the learning process. If you make a mistake while performing Polynomial Long Division, don’t worry. Just go back, check your work, find where you went wrong, and try again. With practice, you’ll get better and make fewer mistakes.

Can Polynomial Long Division help in solving real-world problems?

Absolutely! Polynomial Long Division can be applied to solve a variety of real-world problems, especially those involving mathematical modeling, physics, engineering, and computer science.

Does Brighterly provide resources for practicing Polynomial Long Division?

Yes, at Brighterly, we offer a variety of resources including practice problems, interactive quizzes, and detailed solution guides to help you master Polynomial Long Division. Visit our website to explore these resources.

Jo-ann Caballes

5 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.