Use Synthetic Division To Solve ( X 3 − X 2 − 17 X − 15 ) ÷ ( X − 5 \left(x^3-x^2-17x-15\right) \div (x-5 ( X 3 − X 2 − 17 X − 15 ) ÷ ( X − 5 ]. What Is The Quotient?A. X 2 + 4 X + 3 X^2+4x+3 X 2 + 4 X + 3 B. X 2 − 6 X + 13 − 80 X − 5 X^2-6x+13-\frac{80}{x-5} X 2 − 6 X + 13 − X − 5 80 ​ C. X 3 + 4 X 2 + 3 X X^3+4x^2+3x X 3 + 4 X 2 + 3 X D. X 2 − 6 X + 13 − 80 X + 5 X^2-6x+13-\frac{80}{x+5} X 2 − 6 X + 13 − X + 5 80 ​

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a). In this article, we will use synthetic division to solve the problem $\left(x^3-x^2-17x-15\right) \div (x-5)$ and find the quotient.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a). The method involves using a single row of numbers to perform the division, rather than the multiple rows required by the long division method.

How to Perform Synthetic Division

To perform synthetic division, we need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the factor (x - a) that we are dividing by.
3. Bring down the first coefficient.
4. Multiply the value of the factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Multiply the value of the factor by the result from step 5 and write the result below the third coefficient.
7. Add the third coefficient and the result from step 6.
8. Repeat steps 6 and 7 until we have added all the coefficients.
9. The final result is the quotient.

Solving the Problem

Now that we have learned how to perform synthetic division, let's use it to solve the problem $\left(x^3-x^2-17x-15\right) \div (x-5)$.

Step 1: Write Down the Coefficients

The coefficients of the polynomial are 1, -1, -17, and -15. We will write them down in a row, with the constant term on the right.

| 1 | -1 | -17 | -15 |

Step 2: Write Down the Value of the Factor

The value of the factor (x - 5) is 5.

Step 3: Bring Down the First Coefficient

We will bring down the first coefficient, which is 1.

| 1 | -1 | -17 | -15 | | 5 | | | |

Step 4: Multiply the Value of the Factor by the First Coefficient

We will multiply the value of the factor (5) by the first coefficient (1) and write the result below the second coefficient.

| 1 | -1 | -17 | -15 | | 5 | 5 | | |

Step 5: Add the Second Coefficient and the Result

We will add the second coefficient (-1) and the result from step 4 (5) to get 4.

| 1 | -1 | -17 | -15 | | 5 | 4 | | |

Step 6: Multiply the Value of the Factor by the Result

We will multiply the value of the factor (5) by the result from step 5 (4) to get 20.

| 1 | -1 | -17 | -15 | | 5 | 4 | 20 | |

Step 7: Add the Third Coefficient and the Result

We will add the third coefficient (-17) and the result from step 6 (20) to get 3.

| 1 | -1 | -17 | -15 | | 5 | 4 | 3 | |

Step 8: Multiply the Value of the Factor by the Result

We will multiply the value of the factor (5) by the result from step 7 (3) to get 15.

| 1 | -1 | -17 | -15 | | 5 | 4 | 3 | 15 |

Step 9: Add the Fourth Coefficient and the Result

We will add the fourth coefficient (-15) and the result from step 8 (15) to get 0.

| 1 | -1 | -17 | -15 | | 5 | 4 | 3 | 0 |

The Quotient

The final result is the quotient, which is $x^2+4x+3$.

Conclusion

In this article, we used synthetic division to solve the problem $\left(x^3-x^2-17x-15\right) \div (x-5)$ and found the quotient to be $x^2+4x+3$. Synthetic division is a powerful tool for polynomial division and is particularly useful when dividing polynomials by factors of the form (x - a). We hope that this article has helped you to understand how to use synthetic division to solve polynomial division problems.

References

• [1] "Synthetic Division" by Math Open Reference. Retrieved from https://www.mathopenref.com/syntheticdivision.html
• [2] "Polynomial Division" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7d7/x2f4f7d7-step-1/x2f4f7d7-step-2

Discussion

What is synthetic division? How does it work? What are the steps involved in performing synthetic division? Have you ever used synthetic division to solve a polynomial division problem? If so, please share your experience with us.

Introduction

Synthetic division is a powerful tool for polynomial division, but it can be a bit confusing at first. In this article, we will answer some of the most frequently asked questions about synthetic division, including what it is, how it works, and how to use it to solve polynomial division problems.

Q: What is Synthetic Division?

A: Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a).

Q: How Does Synthetic Division Work?

A: Synthetic division works by using a single row of numbers to perform the division, rather than the multiple rows required by the long division method. The method involves bringing down the first coefficient, multiplying the value of the factor by the first coefficient, adding the second coefficient and the result, and repeating the process until all the coefficients have been added.

Q: What Are the Steps Involved in Performing Synthetic Division?

A: The steps involved in performing synthetic division are:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the factor (x - a) that we are dividing by.
3. Bring down the first coefficient.
4. Multiply the value of the factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Multiply the value of the factor by the result from step 5 and write the result below the third coefficient.
7. Add the third coefficient and the result from step 6.
8. Repeat steps 6 and 7 until we have added all the coefficients.
9. The final result is the quotient.

Q: What Is the Quotient in Synthetic Division?

A: The quotient in synthetic division is the result of the division, which is the polynomial that we are dividing by.

Q: How Do I Use Synthetic Division to Solve Polynomial Division Problems?

A: To use synthetic division to solve polynomial division problems, follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the factor (x - a) that we are dividing by.
3. Bring down the first coefficient.
4. Multiply the value of the factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Multiply the value of the factor by the result from step 5 and write the result below the third coefficient.
7. Add the third coefficient and the result from step 6.
8. Repeat steps 6 and 7 until we have added all the coefficients.
9. The final result is the quotient.

Q: What Are Some Common Mistakes to Avoid When Using Synthetic Division?

A: Some common mistakes to avoid when using synthetic division include:

• Not writing down the coefficients of the polynomial in the correct order.
• Not writing down the value of the factor (x - a) that we are dividing by.
• Not bringing down the first coefficient.
• Not multiplying the value of the factor by the first coefficient and writing the result below the second coefficient.
• Not adding the second coefficient and the result from step 4.
• Not repeating steps 6 and 7 until we have added all the coefficients.

Q: How Do I Check My Work When Using Synthetic Division?

A: To check your work when using synthetic division, follow these steps:

1. Write down the quotient and the remainder.
2. Multiply the quotient by the factor (x - a) and add the remainder.
3. If the result is equal to the original polynomial, then your work is correct.

Conclusion

Synthetic division is a powerful tool for polynomial division, but it can be a bit confusing at first. By following the steps outlined in this article, you should be able to use synthetic division to solve polynomial division problems with ease. Remember to avoid common mistakes and to check your work carefully.

References

• [1] "Synthetic Division" by Math Open Reference. Retrieved from https://www.mathopenref.com/syntheticdivision.html
• [2] "Polynomial Division" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7d7/x2f4f7d7-step-1/x2f4f7d7-step-2

Discussion

Do you have any questions about synthetic division? Have you ever used synthetic division to solve a polynomial division problem? If so, please share your experience with us.