Identify The Polynomial Divisor, Dividend, And Quotient Represented By The Synthetic Division.$\[ -3 \left\lvert\, \begin{array}{rrrrr} 2 & 11 & 18 & 9 \\ & -6 & -15 & -9 \\ \hline 2 & 5 & 3 & 0 \end{array}\right. \\]Divisor: A.

by ADMIN 229 views

Introduction

Synthetic division is a mathematical technique used to divide polynomials. It is a simplified method of polynomial long division that is particularly useful for dividing polynomials by linear factors. In this article, we will explore the concept of synthetic division and how to identify the polynomial divisor, dividend, and quotient represented by the synthetic division.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials that involves a series of steps. It is a shortcut to polynomial long division and is particularly useful for dividing polynomials by linear factors. The process of synthetic division involves dividing the polynomial by a linear factor of the form (x - a), where a is a constant.

The Synthetic Division Process

The synthetic division process involves the following steps:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
  2. Bring down the first coefficient.
  3. Multiply the number at the bottom of the column by the number at the top of the column, and write the result below the next coefficient.
  4. Add the numbers in the second column, and write the result below the line.
  5. Repeat steps 3 and 4 until you reach the last coefficient.
  6. The final result is the quotient and remainder.

Identifying the Polynomial Divisor, Dividend, and Quotient

In the given example, we have the following synthetic division:

{ -3 \left\lvert\, \begin{array}{rrrrr} 2 & 11 & 18 & 9 \\ & -6 & -15 & -9 \\ \hline 2 & 5 & 3 & 0 \end{array}\right. \}

To identify the polynomial divisor, dividend, and quotient, we need to analyze the synthetic division.

  • Divisor: The divisor is the linear factor that we are dividing the polynomial by. In this case, the divisor is (x + 3).
  • Dividend: The dividend is the polynomial that we are dividing. In this case, the dividend is 2x^3 + 11x^2 + 18x + 9.
  • Quotient: The quotient is the result of dividing the dividend by the divisor. In this case, the quotient is 2x^2 + 5x + 3.

Step-by-Step Solution

Let's break down the solution step by step:

  1. Step 1: Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.

    {

\begin{array}{rrrrr} 2 & 11 & 18 & 9 \ \end{array} }$

  1. Step 2: Bring down the first coefficient.

    {

\begin{array}{rrrrr} 2 & 11 & 18 & 9 \ & 2 \ \hline 2 & 5 & 3 & 0 \end{array} }$

  1. Step 3: Multiply the number at the bottom of the column by the number at the top of the column, and write the result below the next coefficient.

    {

\begin{array}{rrrrr} 2 & 11 & 18 & 9 \ & -6 & -15 & -9 \ \hline 2 & 5 & 3 & 0 \end{array} }$

  1. Step 4: Add the numbers in the second column, and write the result below the line.

    {

\begin{array}{rrrrr} 2 & 11 & 18 & 9 \ & -6 & -15 & -9 \ \hline 2 & 5 & 3 & 0 \end{array} }$

  1. Step 5: Repeat steps 3 and 4 until you reach the last coefficient.

    {

\begin{array}{rrrrr} 2 & 11 & 18 & 9 \ & -6 & -15 & -9 \ \hline 2 & 5 & 3 & 0 \end{array} }$

Conclusion

In this article, we have explored the concept of synthetic division and how to identify the polynomial divisor, dividend, and quotient represented by the synthetic division. We have also provided a step-by-step solution to the given example. Synthetic division is a powerful tool for dividing polynomials, and it is an essential concept in algebra.

Frequently Asked Questions

  • What is synthetic division?

    Synthetic division is a method of dividing polynomials that involves a series of steps. It is a shortcut to polynomial long division and is particularly useful for dividing polynomials by linear factors.

  • How do I identify the polynomial divisor, dividend, and quotient?

    To identify the polynomial divisor, dividend, and quotient, you need to analyze the synthetic division. The divisor is the linear factor that we are dividing the polynomial by, the dividend is the polynomial that we are dividing, and the quotient is the result of dividing the dividend by the divisor.

  • What are the steps involved in synthetic division?

    The steps involved in synthetic division are:

    1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
    2. Bring down the first coefficient.
    3. Multiply the number at the bottom of the column by the number at the top of the column, and write the result below the next coefficient.
    4. Add the numbers in the second column, and write the result below the line.
    5. Repeat steps 3 and 4 until you reach the last coefficient.

References

  • Algebra: A Comprehensive Introduction by Michael Artin
  • Polynomial Division: A Guide to Synthetic Division by James Stewart
  • Mathematics: A Guide to Synthetic Division by David C. Lay

Further Reading

  • Synthetic Division: A Guide to Dividing Polynomials by Khan Academy
  • Polynomial Division: A Guide to Synthetic Division by MIT OpenCourseWare
  • Mathematics: A Guide to Synthetic Division by Wolfram Alpha
    Synthetic Division: A Comprehensive Guide to Identifying Polynomial Divisor, Dividend, and Quotient ===========================================================

Q&A: Synthetic Division

Q: What is synthetic division?

A: Synthetic division is a method of dividing polynomials that involves a series of steps. It is a shortcut to polynomial long division and is particularly useful for dividing polynomials by linear factors.

Q: How do I identify the polynomial divisor, dividend, and quotient?

A: To identify the polynomial divisor, dividend, and quotient, you need to analyze the synthetic division. The divisor is the linear factor that we are dividing the polynomial by, the dividend is the polynomial that we are dividing, and the quotient is the result of dividing the dividend by the divisor.

Q: What are the steps involved in synthetic division?

A: The steps involved in synthetic division are:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
  2. Bring down the first coefficient.
  3. Multiply the number at the bottom of the column by the number at the top of the column, and write the result below the next coefficient.
  4. Add the numbers in the second column, and write the result below the line.
  5. Repeat steps 3 and 4 until you reach the last coefficient.

Q: What is the difference between synthetic division and polynomial long division?

A: Synthetic division is a shortcut to polynomial long division. It involves a series of steps that are similar to polynomial long division, but it is faster and more efficient.

Q: Can I use synthetic division to divide polynomials by quadratic factors?

A: No, synthetic division is only used to divide polynomials by linear factors. If you need to divide a polynomial by a quadratic factor, you will need to use polynomial long division.

Q: How do I know if a polynomial can be divided by a linear factor?

A: A polynomial can be divided by a linear factor if the remainder is zero. If the remainder is not zero, then the polynomial cannot be divided by the linear factor.

Q: Can I use synthetic division to divide polynomials with complex coefficients?

A: Yes, synthetic division can be used to divide polynomials with complex coefficients. However, you will need to use complex numbers and follow the same steps as you would with real numbers.

Q: How do I handle negative coefficients in synthetic division?

A: When you have a negative coefficient in synthetic division, you will need to multiply the number at the bottom of the column by the negative number at the top of the column. This will give you a positive result.

Q: Can I use synthetic division to divide polynomials with fractional coefficients?

A: Yes, synthetic division can be used to divide polynomials with fractional coefficients. However, you will need to follow the same steps as you would with real numbers, and you will need to be careful when multiplying and adding fractions.

Q: How do I know if a polynomial is divisible by a linear factor?

A: A polynomial is divisible by a linear factor if the remainder is zero. If the remainder is not zero, then the polynomial is not divisible by the linear factor.

Q: Can I use synthetic division to divide polynomials with multiple linear factors?

A: Yes, synthetic division can be used to divide polynomials with multiple linear factors. However, you will need to follow the same steps as you would with a single linear factor, and you will need to be careful when multiplying and adding the results.

Q: How do I handle repeated linear factors in synthetic division?

A: When you have a repeated linear factor in synthetic division, you will need to multiply the number at the bottom of the column by the repeated number at the top of the column. This will give you a positive result.

Q: Can I use synthetic division to divide polynomials with irrational coefficients?

A: Yes, synthetic division can be used to divide polynomials with irrational coefficients. However, you will need to follow the same steps as you would with real numbers, and you will need to be careful when multiplying and adding the results.

Q: How do I know if a polynomial is divisible by a linear factor with a complex coefficient?

A: A polynomial is divisible by a linear factor with a complex coefficient if the remainder is zero. If the remainder is not zero, then the polynomial is not divisible by the linear factor.

Q: Can I use synthetic division to divide polynomials with multiple complex coefficients?

A: Yes, synthetic division can be used to divide polynomials with multiple complex coefficients. However, you will need to follow the same steps as you would with real numbers, and you will need to be careful when multiplying and adding the results.

Conclusion

In this article, we have provided a comprehensive guide to synthetic division, including the steps involved, how to identify the polynomial divisor, dividend, and quotient, and how to handle different types of coefficients. We have also provided a list of frequently asked questions and answers to help you better understand synthetic division.

Frequently Asked Questions

  • What is synthetic division?

    Synthetic division is a method of dividing polynomials that involves a series of steps. It is a shortcut to polynomial long division and is particularly useful for dividing polynomials by linear factors.

  • How do I identify the polynomial divisor, dividend, and quotient?

    To identify the polynomial divisor, dividend, and quotient, you need to analyze the synthetic division. The divisor is the linear factor that we are dividing the polynomial by, the dividend is the polynomial that we are dividing, and the quotient is the result of dividing the dividend by the divisor.

  • What are the steps involved in synthetic division?

    The steps involved in synthetic division are:

    1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
    2. Bring down the first coefficient.
    3. Multiply the number at the bottom of the column by the number at the top of the column, and write the result below the next coefficient.
    4. Add the numbers in the second column, and write the result below the line.
    5. Repeat steps 3 and 4 until you reach the last coefficient.

References

  • Algebra: A Comprehensive Introduction by Michael Artin
  • Polynomial Division: A Guide to Synthetic Division by James Stewart
  • Mathematics: A Guide to Synthetic Division by David C. Lay

Further Reading

  • Synthetic Division: A Guide to Dividing Polynomials by Khan Academy
  • Polynomial Division: A Guide to Synthetic Division by MIT OpenCourseWare
  • Mathematics: A Guide to Synthetic Division by Wolfram Alpha