Division of Polynomials (Long Division of Polynomials)

D&E's Edu Corner

16 Sept 202407:41

Summary

TLDRThe video demonstrates how to solve a polynomial division problem using long division. It walks through dividing the polynomial 2x³ + 3x² - 5x + 6 by x - 2. The process includes dividing the leading terms, multiplying, subtracting, and repeating until reaching the remainder. The final quotient is 2x² + 7x + 9, with a remainder of 24, which is expressed as a fraction over the divisor. The solution is presented as 2x² + 7x + 9 + 24/(x-2).

Takeaways

📝 The problem involves solving polynomial division using the long division method.
🔢 The dividend is 2x³ + 3x² - 5x + 6, and the divisor is x - 2.
➗ The first step is dividing 2x³ by x, resulting in 2x².
🔄 Multiplying 2x² by (x - 2) gives 2x³ - 4x², and subtracting it from the dividend results in 7x².
🔢 The next step is dividing 7x² by x, which gives 7x.
🔄 Multiplying 7x by (x - 2) gives 7x² - 14x, and subtracting it from the current terms results in 9x.
➕ Dividing 9x by x results in 9, which is then multiplied by (x - 2), yielding 9x - 18.
🔄 Subtracting 9x - 18 from the current terms results in a remainder of 24.
🧮 The quotient from the division is 2x² + 7x + 9, and the remainder is 24.
✅ The final result is expressed as 2x² + 7x + 9 + 24/(x - 2).

Q & A

What is the given polynomial division problem in the script?
-The problem is to divide the polynomial 2x³ + 3x² - 5x + 6 by the divisor x - 2 using long division.
What is the first step in polynomial long division according to the script?
-The first step is to divide the leading term of the dividend (2x³) by the leading term of the divisor (x), which results in 2x².
How do you multiply the first quotient term by the divisor?
-You multiply 2x² by x - 2, which gives 2x³ - 4x².
What happens after subtracting the first multiplication result?
-After subtracting 2x³ - 4x² from the original terms, you get 0 for the x³ term and 7x² after simplifying the second term.
What is done after bringing down the next term?
-After bringing down -5x, the next step is to divide the new leading term (7x²) by the leading term of the divisor (x), which gives 7x.
What is the result after multiplying the second quotient term by the divisor?
-Multiplying 7x by x - 2 results in 7x² - 14x.
How is the subtraction carried out in the second step of division?
-After subtracting 7x² - 14x from the previous terms, you get 9x by simplifying -5x + 14x.
What is the quotient after the third division?
-Dividing 9x by x results in the next quotient term, which is 9.
How is the final remainder calculated?
-Multiplying 9 by x - 2 results in 9x - 18. After subtracting, the remainder becomes 24.
What is the final answer in polynomial long division?
-The final answer is 2x² + 7x + 9 with a remainder of 24, which can be written as 2x² + 7x + 9 + 24/(x - 2).

Outlines

00:00

🔢 Polynomial Long Division Step-by-Step

In this segment, the presenter explains how to solve a polynomial division problem using the long division method. The dividend is the polynomial \(2x^3 + 3x^2 - 5x + 6\) and the divisor is \(x - 2\). The presenter walks through the process of dividing the leading term of the dividend by the leading term of the divisor, which results in \(2x^2\). This is then multiplied by the divisor and subtracted from the original dividend, simplifying the polynomial step-by-step. Each operation is clearly shown, including the handling of signs during subtraction, and the intermediate results are carefully calculated.

05:03

✅ Final Calculation and Result Presentation

The presenter continues the long division process, dividing the simplified polynomial \(7x^2 - 5x\) by the divisor's leading term to get \(7x\). This value is multiplied back and subtracted again, resulting in a new polynomial. The process is repeated with the remaining terms, leading to the final quotient \(2x^2 + 7x + 9\) and a remainder of 24. The final answer is expressed as the quotient plus the remainder over the divisor, resulting in the complete solution: \(2x^2 + 7x + 9 + \frac{24}{x-2}\). The presenter highlights the structure of the solution, ensuring clarity in understanding how each component fits into the final answer.

Mindmap

Keywords

💡Long Division of Polynomials

Long Division of Polynomials is a method used to divide one polynomial by another. It is similar to the long division process used in arithmetic but is applied to polynomials. In the video, this method is used to divide the polynomial 2x^3 + 3x^2 - 5x + 6 by x - 2. The process involves dividing the leading term of the dividend by the leading term of the divisor, then multiplying the divisor by this result and subtracting it from the dividend, and repeating this process with the remainder until the degree of the remainder is less than the divisor.

💡Dividend

The Dividend is the polynomial that is being divided in a polynomial division. In the context of the video, the dividend is the polynomial 2x^3 + 3x^2 - 5x + 6. It is the expression inside the long division symbol that we are breaking down using the division process.

💡Divisor

The Divisor is the polynomial by which the dividend is divided. In the video's script, the divisor is x - 2. It is the polynomial outside the long division symbol that is used to divide into the dividend.

💡Quotient

The Quotient is the result of the division of two polynomials. It is the polynomial that you get before considering the remainder. In the video, the quotient is 2x^2 + 7x + 9, which is the result of dividing the dividend by the divisor and is part of the final answer.

💡Remainder

The Remainder is what is left over after the division process. It is the part of the dividend that is not completely divisible by the divisor. In the script, the remainder is 24, which is the value that, when divided by the divisor x - 2, does not result in a whole number.

💡Leading Term

The Leading Term is the term with the highest degree in a polynomial. In the division process, you divide the leading term of the dividend by the leading term of the divisor. For example, in the dividend 2x^3 + 3x^2 - 5x + 6, the leading term is 2x^3, and in the divisor x - 2, it is x.

💡Degree

The Degree of a polynomial is the highest power of the variable in the polynomial. It is important in polynomial division because the degree of the remainder must be less than the degree of the divisor. In the video, the degrees of the dividend and divisor are considered to ensure the division process is carried out correctly.

💡Multiplication

Multiplication is used in the division process to multiply the entire divisor by the result of the division of the leading terms. In the script, after finding that 2x^2 is the result of the initial division, the divisor x - 2 is multiplied by 2x^2 to be subtracted from the dividend.

💡Subtraction

Subtraction is used in polynomial division to subtract the product of the divisor and the quotient term from the current polynomial remainder. This is done to find a new remainder to continue the division process. In the script, after multiplying, subtraction is used to get the new remainder.

💡Polynomial

A Polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. The video's main theme revolves around dividing one polynomial by another, showcasing the steps and calculations involved.

💡Variable

A Variable is a symbol, usually a letter, that represents an unknown value in a polynomial. In the video, the variable is x, and it is used in both the dividend 2x^3 + 3x^2 - 5x + 6 and the divisor x - 2. Variables allow for general solutions in algebra.

Highlights

Introduction of the long division method for polynomials.

The dividend in the problem is 2x³ + 3x² - 5x + 6.

The divisor in the problem is x - 2.

Start of the long division process by dividing 2x³ by x, resulting in 2x².

Multiplying 2x² by (x - 2), resulting in 2x³ - 4x².

Subtraction of 2x³ - 4x² from the original polynomial, yielding 7x².

Bringing down -5x to get 7x² - 5x.

Dividing 7x² by x to get 7x.

Multiplying 7x by (x - 2), resulting in 7x² - 14x.

Subtracting 7x² - 14x from 7x² - 5x, yielding 9x.

Bringing down the constant term 6, leading to the expression 9x + 6.

Dividing 9x by x to get 9.

Multiplying 9 by (x - 2), resulting in 9x - 18.

Final subtraction yielding the remainder 24.

The final quotient is 2x² + 7x + 9, and the remainder is 24.

The solution is presented in the form of (quotient) + (remainder)/(divisor), which is 2x² + 7x + 9 + 24/(x - 2).