What Dividend Is Represented By The Synthetic Division Below?$\[ \begin{array}{cccc|c} -5 & \vert & 2 & 10 & 1 & 5 \\ & & & -10 & 0 & -5 \\ \hline & & 2 & 0 & 1 & 0 \\ \end{array} \\]A. \[$-10x^2 - 5\$\]B.

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Introduction to Synthetic Division

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus. The process involves dividing the polynomial by a linear factor of the form (x - c), where c is a constant. In this article, we will focus on understanding the synthetic division process and how to determine the dividend represented by the given synthetic division.

Understanding the Synthetic Division Process

The synthetic division process involves dividing the polynomial by a linear factor of the form (x - c). The process is as follows:

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

Analyzing the Given Synthetic Division

The given synthetic division is:

{ \begin{array}{cccc|c} -5 & \vert & 2 & 10 & 1 & 5 \\ & & & -10 & 0 & -5 \\ \hline & & 2 & 0 & 1 & 0 \\ \end{array} \}

Determining the Dividend

To determine the dividend represented by the synthetic division, we need to analyze the numbers in the bottom row. The numbers in the bottom row represent the coefficients of the quotient polynomial. In this case, the numbers in the bottom row are 2, 0, and 1.

Writing the Dividend Polynomial

The dividend polynomial can be written by multiplying the coefficients in the bottom row by the corresponding powers of x. In this case, the dividend polynomial is:

2x^2 + 0x + 1

Simplifying the Dividend Polynomial

The dividend polynomial can be simplified by combining like terms. In this case, the dividend polynomial is already simplified.

Conclusion

In conclusion, the dividend represented by the synthetic division below is 2x^2 + 0x + 1, which can be simplified to 2x^2 + 1.

Final Answer

The final answer is 2x2+1\boxed{2x^2 + 1}.

Discussion

The given synthetic division represents the division of a polynomial by a linear factor of the form (x - c). The dividend represented by the synthetic division is the quotient polynomial, which is 2x^2 + 1. This polynomial can be used to represent a variety of functions, including quadratic functions.

Related Topics

Synthetic division is a powerful tool used in algebra and calculus to divide polynomials by linear factors. It is a shortcut to the long division method and is often used to find the quotient and remainder of a polynomial division. Some related topics include:

  • Long division of polynomials
  • Factoring polynomials
  • Quadratic functions
  • Algebraic expressions

Example Problems

Here are some example problems that demonstrate the use of synthetic division:

  • Divide the polynomial 3x^2 + 2x + 1 by the linear factor (x + 1).
  • Divide the polynomial 2x^2 - 3x + 1 by the linear factor (x - 2).
  • Divide the polynomial x^2 + 4x + 4 by the linear factor (x + 2).

Practice Problems

Here are some practice problems that demonstrate the use of synthetic division:

  • Divide the polynomial 4x^2 + 3x + 2 by the linear factor (x + 1).
  • Divide the polynomial 3x^2 - 2x + 1 by the linear factor (x - 3).
  • Divide the polynomial 2x^2 + 4x + 4 by the linear factor (x + 2).

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • Divide the polynomial 4x^2 + 3x + 2 by the linear factor (x + 1): 4x + 1
  • Divide the polynomial 3x^2 - 2x + 1 by the linear factor (x - 3): 3x + 10
  • Divide the polynomial 2x^2 + 4x + 4 by the linear factor (x + 2): 2x + 0

Introduction

Synthetic division is a powerful tool used in algebra and calculus to divide polynomials by linear factors. It is a shortcut to the long division method and is often used to find the quotient and remainder of a polynomial division. In this article, we will answer some frequently asked questions about synthetic division and dividend polynomials.

Q1: What is synthetic division?

A1: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus.

Q2: How do I perform synthetic division?

A2: To perform synthetic division, you need to follow these steps:

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

Q3: What is the dividend polynomial?

A3: The dividend polynomial is the polynomial that is being divided by the linear factor. It is the polynomial that is represented by the coefficients in the bottom row of the synthetic division table.

Q4: How do I write the dividend polynomial?

A4: To write the dividend polynomial, you need to multiply the coefficients in the bottom row by the corresponding powers of x. For example, if the coefficients in the bottom row are 2, 0, and 1, the dividend polynomial is 2x^2 + 0x + 1.

Q5: Can I use synthetic division to divide polynomials by quadratic factors?

A5: No, synthetic division can only be used to divide polynomials by linear factors. If you need to divide a polynomial by a quadratic factor, you will need to use a different method, such as long division or factoring.

Q6: What is the remainder in synthetic division?

A6: The remainder in synthetic division is the final number in the bottom row of the synthetic division table. It represents the amount left over after dividing the polynomial by the linear factor.

Q7: Can I use synthetic division to find the quotient and remainder of a polynomial division?

A7: Yes, synthetic division can be used to find the quotient and remainder of a polynomial division. The quotient is the polynomial that is obtained by dividing the dividend polynomial by the linear factor, and the remainder is the amount left over.

Q8: How do I use synthetic division to find the roots of a polynomial?

A8: Synthetic division can be used to find the roots of a polynomial by dividing the polynomial by a linear factor and checking if the remainder is zero. If the remainder is zero, then the linear factor is a root of the polynomial.

Q9: Can I use synthetic division to divide polynomials with complex coefficients?

A9: Yes, synthetic division can be used to divide polynomials with complex coefficients. The process is the same as for polynomials with real coefficients, but you will need to use complex numbers to represent the coefficients.

Q10: Are there any limitations to using synthetic division?

A10: Yes, synthetic division can only be used to divide polynomials by linear factors. If you need to divide a polynomial by a quadratic factor or a higher-degree factor, you will need to use a different method.

Conclusion

Synthetic division is a powerful tool used in algebra and calculus to divide polynomials by linear factors. It is a shortcut to the long division method and is often used to find the quotient and remainder of a polynomial division. In this article, we have answered some frequently asked questions about synthetic division and dividend polynomials.

Final Answer

The final answer is that synthetic division is a useful tool for dividing polynomials by linear factors, but it has limitations and can only be used for polynomials with real or complex coefficients.

Related Topics

Synthetic division is a related topic to:

  • Long division of polynomials
  • Factoring polynomials
  • Quadratic functions
  • Algebraic expressions

Example Problems

Here are some example problems that demonstrate the use of synthetic division:

  • Divide the polynomial 3x^2 + 2x + 1 by the linear factor (x + 1).
  • Divide the polynomial 2x^2 - 3x + 1 by the linear factor (x - 2).
  • Divide the polynomial x^2 + 4x + 4 by the linear factor (x + 2).

Practice Problems

Here are some practice problems that demonstrate the use of synthetic division:

  • Divide the polynomial 4x^2 + 3x + 2 by the linear factor (x + 1).
  • Divide the polynomial 3x^2 - 2x + 1 by the linear factor (x - 3).
  • Divide the polynomial 2x^2 + 4x + 4 by the linear factor (x + 2).

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • Divide the polynomial 4x^2 + 3x + 2 by the linear factor (x + 1): 4x + 1
  • Divide the polynomial 3x^2 - 2x + 1 by the linear factor (x - 3): 3x + 10
  • Divide the polynomial 2x^2 + 4x + 4 by the linear factor (x + 2): 2x + 0