How to Divide Polynomials using Long Division - Polynomials

MATH TEACHER GON

17 Sept 202211:43

Summary

TLDRIn this educational video, Trigon demonstrates how to divide polynomials using long division. Starting with dividing 6x^2 - 2x - 28 by 2x + 4, Trigon explains the process step-by-step, emphasizing that it mirrors the long division method used for whole numbers. The video illustrates dividing the leading terms, multiplying, and subtracting to find the partial quotient, which in this case is 3x - 7. Trigon then solves another polynomial division problem, reinforcing the method with a new example. The video is designed to teach viewers the algebraic process in an accessible manner.

Takeaways

📚 The video tutorial focuses on dividing polynomials using long division, a method similar to dividing whole numbers but applied to algebraic expressions.
🔢 The first example involves dividing (6x^2 - 2x - 28) by (2x + 4), where the leading terms are divided first, followed by multiplication and subtraction steps.
👨‍🏫 The instructor emphasizes the importance of dividing the leading coefficients or terms of the dividend and divisor to find the partial quotient.
📉 After dividing, the instructor demonstrates how to multiply the entire divisor by the partial quotient and subtract this product from the original polynomial to find the remainder.
🔄 The process is repeated with the new polynomial (result after subtraction) to find subsequent terms of the quotient until the remainder is zero or less than the divisor.
📝 The quotient obtained from the division is (3x - 7), indicating that the division process has been correctly followed and the remainder is zero.
🔍 In a second problem, the method is applied to (3x^3 - 4x^2 - 7x - 5) divided by (3x - 2), showing the consistency of the long division approach.
📈 The tutorial highlights the need to bring down terms from the dividend when the current degree of the remainder is less than the degree of the divisor.
📌 The final quotient for the second problem is (x^2 + 2x - 1) with a remainder of (-7), which is expressed as -7/(3x - 2).
💡 The video concludes with a reminder to like, subscribe, and enable notifications for updates, encouraging viewer engagement with the channel.

Q & A

What is the main topic discussed in the video?
-The main topic discussed in the video is dividing polynomials using long division.
What is the first polynomial division problem presented in the video?
-The first polynomial division problem is 6x^2 - 2x - 28 divided by 2x + 4.
How does the video explain the process of polynomial long division?
-The video explains that polynomial long division is similar to the long division of whole numbers, but in algebraic form. It involves dividing the leading terms, multiplying, and then subtracting.
What is the partial quotient obtained after the first division step in the first problem?
-The partial quotient obtained after the first division step in the first problem is 3x.
What is the result of the multiplication of the partial quotient 3x by the divisor 2x + 4?
-The result of the multiplication of the partial quotient 3x by the divisor 2x + 4 is 6x^2 + 12x.
What is the remainder after the first subtraction step in the first problem?
-The remainder after the first subtraction step in the first problem is -14x - 28.
What is the final quotient obtained after solving the first polynomial division problem?
-The final quotient obtained after solving the first polynomial division problem is 3x - 7.
How does the video handle the remainder in the division process?
-The video brings down the remainder and continues the division process by dividing the leading term of the new polynomial by the leading term of the divisor, then multiplying and subtracting as before.
What is the second polynomial division problem presented in the video?
-The second polynomial division problem is 3x^3 - 4x^2 - 7x - 5 divided by 3x - 2.
What is the final quotient and remainder obtained after solving the second polynomial division problem?
-The final quotient obtained after solving the second polynomial division problem is x^2 + 2x, and the remainder is -7/(3x - 2).
What is the advice given by the video for those who are new to the channel?
-The video advises those who are new to the channel to like, subscribe, and hit the Bell button to be updated with the latest uploads.

Outlines

00:00

📘 Polynomial Long Division Introduction

The video begins with the host, Trigon, introducing the topic of dividing polynomials using long division. The first problem presented is dividing \(6x^2 - 2x - 28\) by \(2x + 4\). The process involves dividing the leading terms of the dividend and divisor, which results in \(3x\) as the partial quotient. The host then demonstrates the steps of multiplying the divisor by the partial quotient and subtracting the result from the dividend. The process is repeated until the remainder is zero, yielding the quotient \(3x - 7\).

05:01

📗 Detailed Polynomial Long Division Process

The second paragraph delves deeper into the long division process with a new problem: dividing \(3x^3 - 2x^2 - 7x - 5\) by \(3x - 2\). The host explains how to divide the leading terms, resulting in \(x^2\) as the partial quotient. The video shows the multiplication of the divisor by the partial quotient and the subsequent subtraction from the dividend. The process is repeated, leading to a remainder of \(-7\) over the divisor \(3x - 2\), and the quotient is \(x^2 + 2x - 1\).

10:04

📙 Conclusion and Call to Action

In the final paragraph, the host concludes the tutorial on polynomial long division by summarizing the steps and presenting the final answer for the second problem: \(x^2 + 2x - 1\) with a remainder of \(-7/(3x - 2)\). The host encourages viewers to like, subscribe, and turn on notifications for updates, and signs off as 'Teacher Trigon'.

Mindmap

Keywords

💡Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. In the video, polynomials are the main objects of study, specifically for division using long division. The script mentions dividing '6X Square minus 2x minus 28' by '2x plus 4,' which are both polynomials.

💡Long Division

Long division is a method used for dividing larger numbers or expressions, such as polynomials. In the video, the presenter explains how to divide polynomials using the long division technique, which is analogous to the method used for whole numbers but adapted for algebraic expressions. The script walks through the process of dividing '6X Square minus 2x minus 28' by '2x plus 4' using long division.

💡Dividend

The dividend is the number or expression that is to be divided in a mathematical operation. In the context of the video, '6X Square minus 2x minus 28' is referred to as the dividend, which is the polynomial being divided by another polynomial, the divisor.

💡Divisor

The divisor is the number or expression by which the dividend is divided. In the video script, '2x plus 4' is the divisor, which is used to divide the dividend '6X Square minus 2x minus 28'. The process involves dividing the leading terms and then performing multiplication and subtraction steps.

💡Leading Coefficients

Leading coefficients are the numerical factors that multiply the highest power of the variable in a term of a polynomial. The video explains that the first step in dividing polynomials is to divide the leading coefficients of the dividend and divisor, which in the example is '6' divided by '2' from the terms '6X Square' and '2x', respectively.

💡Partial Quotient

A partial quotient is the result of dividing the leading terms of the dividend by the divisor in a step of the long division process. In the video, after dividing the leading terms, '3x' is identified as the partial quotient, which is then used in subsequent multiplication and subtraction steps.

💡Multiply and Subtract

This phrase describes two of the key steps in the long division process for polynomials. After obtaining the partial quotient, the video shows how to multiply it by the entire divisor and then subtract the result from the dividend to get a new expression to continue the division process.

💡Remainder

The remainder is what is left over after the division process when the dividend cannot be divided evenly by the divisor. In the video, the presenter ensures that the remainder is zero, indicating that the division has been carried out correctly and completely for the given polynomials.

💡Quotient

The quotient is the result of a division operation. In the context of the video, after performing the long division, the quotient is '3x minus seven' for the first polynomial division example. This represents the simplified form of the division of the two polynomials.

💡Algebraic Form

Algebraic form refers to the way mathematical expressions are written using variables and operations. The video emphasizes that dividing polynomials using long division is similar to dividing whole numbers but is done in an algebraic form, involving variables and their exponents.

Highlights

Introduction to dividing polynomials using long division

Explanation of the first problem: 6x^2 - 2x - 28 divided by 2x + 4

Step-by-step guide on dividing the leading terms of the dividend and divisor

Calculation of the partial quotient: 6x^2 divided by 2x equals 3x

Multiplication of the partial quotient by the divisor

Subtraction step in the long division process

Continuation of the division process with the next term

Final quotient derivation: 3x - 7

Emphasis on the remainder being zero in the division

Introduction to the next problem involving 3x^3 divided by 3x

Division of the leading terms resulting in x^2

Multiplication and subtraction steps for the second problem

Derivation of the quotient for the second problem: x^2 + 2x

Explanation of the remainder over the divisor

Final expression of the quotient and remainder

Encouragement for viewers to like, subscribe, and hit the Bell button for updates

Conclusion and sign-off by the presenter