Use Synthetic Division To Perform The Division.$ \frac{-8x^3 + 6x^2 - 9x + 7}{x - 2} }$Simplify Your Answer { \square$ $

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Introduction to Synthetic Division

Synthetic division is a method used to divide polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. This method is particularly useful when dividing polynomials by factors of the form (x - a), where a is a constant. In this article, we will use synthetic division to perform the division of the polynomial ${-8x^3 + 6x^2 - 9x + 7}$ by the linear factor ${x - 2}$.

The Synthetic Division Process

The synthetic division process involves the following steps:

  1. Write down the coefficients of the polynomial: The coefficients of the polynomial are the numbers that multiply the powers of x. In this case, the coefficients are -8, 6, -9, and 7.
  2. Write down the root of the linear factor: The root of the linear factor is the number that is being divided out. In this case, the root is 2.
  3. Bring down the first coefficient: The first coefficient is the first number in the list of coefficients. In this case, the first coefficient is -8.
  4. Multiply the root by the first coefficient: Multiply the root by the first coefficient to get -16.
  5. Add the result to the second coefficient: Add the result to the second coefficient to get 6 - 16 = -10.
  6. Multiply the root by the result: Multiply the root by the result to get -20.
  7. Add the result to the third coefficient: Add the result to the third coefficient to get -9 - 20 = -29.
  8. Multiply the root by the result: Multiply the root by the result to get -58.
  9. Add the result to the fourth coefficient: Add the result to the fourth coefficient to get 7 - 58 = -51.

The Result of the Synthetic Division

The result of the synthetic division is a new polynomial with coefficients -8, -10, -29, and -51. This polynomial can be written as ${-8x^3 - 10x^2 - 29x - 51}$.

Simplifying the Result

To simplify the result, we can factor out the greatest common factor (GCF) of the coefficients. In this case, the GCF is -1. Factoring out the GCF, we get ${-(8x^3 + 10x^2 + 29x + 51)}$.

The Final Result

The final result of the synthetic division is the polynomial ${-(8x^3 + 10x^2 + 29x + 51)}$. This polynomial is the quotient of the original polynomial ${-8x^3 + 6x^2 - 9x + 7}$ divided by the linear factor ${x - 2}$.

Conclusion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we used synthetic division to perform the division of the polynomial ${-8x^3 + 6x^2 - 9x + 7}$ by the linear factor ${x - 2}$. The result of the synthetic division was the polynomial ${-(8x^3 + 10x^2 + 29x + 51)}$. This polynomial is the quotient of the original polynomial divided by the linear factor.

Example Problems

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

Example 1

Divide the polynomial ${3x^3 - 2x^2 + 5x - 1}$ by the linear factor ${x + 1}$.

Solution

  1. Write down the coefficients of the polynomial: 3, -2, 5, and -1.
  2. Write down the root of the linear factor: -1.
  3. Bring down the first coefficient: 3.
  4. Multiply the root by the first coefficient: -3.
  5. Add the result to the second coefficient: -2 - 3 = -5.
  6. Multiply the root by the result: 5.
  7. Add the result to the third coefficient: 5 + 5 = 10.
  8. Multiply the root by the result: -10.
  9. Add the result to the fourth coefficient: -1 - 10 = -11.

The result of the synthetic division is the polynomial ${3x^3 - 5x^2 + 10x - 11}$.

Example 2

Divide the polynomial ${2x^3 + 3x^2 - 4x + 1}$ by the linear factor ${x - 3}$.

Solution

  1. Write down the coefficients of the polynomial: 2, 3, -4, and 1.
  2. Write down the root of the linear factor: 3.
  3. Bring down the first coefficient: 2.
  4. Multiply the root by the first coefficient: 6.
  5. Add the result to the second coefficient: 3 + 6 = 9.
  6. Multiply the root by the result: 27.
  7. Add the result to the third coefficient: -4 + 27 = 23.
  8. Multiply the root by the result: 69.
  9. Add the result to the fourth coefficient: 1 + 69 = 70.

The result of the synthetic division is the polynomial ${2x^3 + 9x^2 + 23x + 70}$.

Tips and Tricks

Here are a few tips and tricks for using synthetic division:

  • Make sure to bring down the first coefficient: This is the most important step in synthetic division. Make sure to bring down the first coefficient before multiplying the root by the first coefficient.
  • Use the correct root: Make sure to use the correct root of the linear factor. If the root is not correct, the result of the synthetic division will be incorrect.
  • Check your work: Make sure to check your work by multiplying the root by the result and adding the result to the next coefficient.
  • Use synthetic division for polynomials with linear factors: Synthetic division is particularly useful for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division.

Conclusion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we used synthetic division to perform the division of the polynomial ${-8x^3 + 6x^2 - 9x + 7}$ by the linear factor ${x - 2}$. The result of the synthetic division was the polynomial ${-(8x^3 + 10x^2 + 29x + 51)}$. This polynomial is the quotient of the original polynomial divided by the linear factor. We also provided example problems and tips and tricks for using synthetic division.

Introduction

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we will answer some of the most frequently asked questions about synthetic division.

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division.

Q: How do I perform synthetic division?

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

  1. Write down the coefficients of the polynomial.
  2. Write down the root of the linear factor.
  3. Bring down the first coefficient.
  4. Multiply the root by the first coefficient.
  5. Add the result to the second coefficient.
  6. Multiply the root by the result.
  7. Add the result to the third coefficient.
  8. Multiply the root by the result.
  9. Add the result to the fourth coefficient.

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

A: Synthetic division is a simplified approach to polynomial division that eliminates the need for long division. Long division is a more complex process that involves dividing polynomials by linear factors, but it is not as efficient as synthetic division.

Q: Can I use synthetic division for polynomials with quadratic factors?

A: No, synthetic division is only used for polynomials with linear factors. If you need to divide a polynomial by a quadratic factor, you will need to use a different method, such as factoring or using the quadratic formula.

Q: How do I know if I have performed synthetic division correctly?

A: To check if you have performed synthetic division correctly, you can multiply the root by the result and add the result to the next coefficient. If the result is correct, then you have performed synthetic division correctly.

Q: Can I use synthetic division for polynomials with complex factors?

A: Yes, synthetic division can be used for polynomials with complex factors. However, you will need to use complex numbers and follow the same steps as before.

Q: How do I use synthetic division to divide a polynomial by a factor of the form (x - a)?

A: To use synthetic division to divide a polynomial by a factor of the form (x - a), you need to follow these steps:

  1. Write down the coefficients of the polynomial.
  2. Write down the root of the linear factor, which is a.
  3. Bring down the first coefficient.
  4. Multiply the root by the first coefficient.
  5. Add the result to the second coefficient.
  6. Multiply the root by the result.
  7. Add the result to the third coefficient.
  8. Multiply the root by the result.
  9. Add the result to the fourth coefficient.

Q: Can I use synthetic division to divide a polynomial by a factor of the form (x + a)?

A: Yes, you can use synthetic division to divide a polynomial by a factor of the form (x + a). The only difference is that you will need to use the negative of the root, which is -a.

Q: How do I use synthetic division to divide a polynomial by a factor of the form (x^2 + ax + b)?

A: To use synthetic division to divide a polynomial by a factor of the form (x^2 + ax + b), you will need to use a different method, such as factoring or using the quadratic formula.

Q: Can I use synthetic division to divide a polynomial by a factor of the form (x^3 + ax^2 + bx + c)?

A: No, synthetic division is only used for polynomials with linear factors. If you need to divide a polynomial by a factor of the form (x^3 + ax^2 + bx + c), you will need to use a different method, such as factoring or using the cubic formula.

Q: How do I know if I have performed synthetic division correctly for a polynomial with complex factors?

A: To check if you have performed synthetic division correctly for a polynomial with complex factors, you can multiply the root by the result and add the result to the next coefficient. If the result is correct, then you have performed synthetic division correctly.

Conclusion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we have answered some of the most frequently asked questions about synthetic division. We hope that this article has been helpful in understanding synthetic division and how to use it to divide polynomials by linear factors.

Tips and Tricks

Here are a few tips and tricks for using synthetic division:

  • Make sure to bring down the first coefficient: This is the most important step in synthetic division. Make sure to bring down the first coefficient before multiplying the root by the first coefficient.
  • Use the correct root: Make sure to use the correct root of the linear factor. If the root is not correct, the result of the synthetic division will be incorrect.
  • Check your work: Make sure to check your work by multiplying the root by the result and adding the result to the next coefficient.
  • Use synthetic division for polynomials with linear factors: Synthetic division is particularly useful for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division.

Example Problems

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

Example 1

Divide the polynomial ${3x^3 - 2x^2 + 5x - 1}$ by the linear factor ${x + 1}$.

Solution

  1. Write down the coefficients of the polynomial: 3, -2, 5, and -1.
  2. Write down the root of the linear factor: -1.
  3. Bring down the first coefficient: 3.
  4. Multiply the root by the first coefficient: -3.
  5. Add the result to the second coefficient: -2 - 3 = -5.
  6. Multiply the root by the result: 5.
  7. Add the result to the third coefficient: 5 + 5 = 10.
  8. Multiply the root by the result: -10.
  9. Add the result to the fourth coefficient: -1 - 10 = -11.

The result of the synthetic division is the polynomial ${3x^3 - 5x^2 + 10x - 11}$.

Example 2

Divide the polynomial ${2x^3 + 3x^2 - 4x + 1}$ by the linear factor ${x - 3}$.

Solution

  1. Write down the coefficients of the polynomial: 2, 3, -4, and 1.
  2. Write down the root of the linear factor: 3.
  3. Bring down the first coefficient: 2.
  4. Multiply the root by the first coefficient: 6.
  5. Add the result to the second coefficient: 3 + 6 = 9.
  6. Multiply the root by the result: 27.
  7. Add the result to the third coefficient: -4 + 27 = 23.
  8. Multiply the root by the result: 69.
  9. Add the result to the fourth coefficient: 1 + 69 = 70.

The result of the synthetic division is the polynomial ${2x^3 + 9x^2 + 23x + 70}$.

Conclusion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we have answered some of the most frequently asked questions about synthetic division. We hope that this article has been helpful in understanding synthetic division and how to use it to divide polynomials by linear factors.