The Steps of Synthetic Division

Oct 13, 2025 by wbrum.org 2 views

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for finding the zeros of a polynomial and can be used to simplify the process of polynomial division. In this article, we will use synthetic division to determine if ${$ x = -5 $}$ and ${$ x = 2 $}$ are zeros of the polynomial ${$ f(x) = x^4 - 5x^3 + 5x^2 + 5x - 6 $}$.

What is Synthetic Division?

Synthetic division is a method of dividing a polynomial by a linear factor of the form ${$ (x - c) $}$. It is a shortcut method that eliminates the need for long division and can be used to find the zeros of a polynomial. The process of synthetic division involves dividing the polynomial by the linear factor and finding the remainder.

The Steps of Synthetic Division

The steps of synthetic division are as follows:

Write down the coefficients of the polynomial in a row.
Write down the value of ${$ c $}$ in the middle of the row.
Multiply the value of ${$ c $}$ by the first coefficient and write the result below the row.
Add the result to the second coefficient and write the result below the row.
Multiply the value of ${$ c $}$ by the result and write the result below the row.
Add the result to the third coefficient and write the result below the row.
Repeat steps 5 and 6 until all coefficients have been used.
The final result is the remainder.

Is ${$ x = -5 $}$ a Zero?

To determine if ${$ x = -5 $}$ is a zero, we will use synthetic division with ${$ c = -5 $}$.

Step 1: Write down the coefficients of the polynomial

	1	-5	5	5	-6

Step 2: Write down the value of ${$ c $}$

	1	-5	5	5	-6
-5

Step 3: Multiply the value of ${$ c $}$ by the first coefficient

	1	-5	5	5	-6
-5	-5

Step 4: Add the result to the second coefficient

	1	-5	5	5	-6
-5	-5	0

Step 5: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
-5	-5	0	25

Step 6: Add the result to the third coefficient

	1	-5	5	5	-6
-5	-5	0	25	30

Step 7: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
-5	-5	0	25	30	-125

Step 8: Add the result to the fourth coefficient

	1	-5	5	5	-6
-5	-5	0	25	30	-125

Step 9: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
-5	-5	0	25	30	-125
					125

Step 10: Add the result to the fifth coefficient

	1	-5	5	5	-6
-5	-5	0	25	30	-125
					125

The final result is ${$ 0 $}$, which means that ${$ x = -5 $}$ is a zero of the polynomial.

Is ${$ x = 2 $}$ a Zero?

To determine if ${$ x = 2 $}$ is a zero, we will use synthetic division with ${$ c = 2 $}$.

Step 1: Write down the coefficients of the polynomial

	1	-5	5	5	-6

Step 2: Write down the value of ${$ c $}$

	1	-5	5	5	-6
2

Step 3: Multiply the value of ${$ c $}$ by the first coefficient

	1	-5	5	5	-6
2	2

Step 4: Add the result to the second coefficient

	1	-5	5	5	-6
2	2	-3

Step 5: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
2	2	-3	-6

Step 6: Add the result to the third coefficient

	1	-5	5	5	-6
2	2	-3	-6	11

Step 7: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
2	2	-3	-6	11	-22

Step 8: Add the result to the fourth coefficient

	1	-5	5	5	-6
2	2	-3	-6	11	-22

Step 9: Multiply the value of ${$ c $}$ by the result

	1	-5	5	5	-6
2	2	-3	-6	11	-22
					44

Step 10: Add the result to the fifth coefficient

	1	-5	5	5	-6
2	2	-3	-6	11	-22
					44

The final result is ${$ 44 $}$, which means that ${$ x = 2 $}$ is not a zero of the polynomial.

Conclusion

$In this article, we used synthetic division to determine if ${$ x = -5 \$}$ and ${$ x = 2 \$}$ are zeros of the polynomial ${$ f(x) = x^4 - 5x^3 + 5x^2 + 5x - 6 \$}$. We found that ${$ x = -5 \$}$ is a zero of the polynomial, but ${$ x = 2 \$}$ is not a zero. Synthetic division is a powerful tool for polynomial division and can be used to simplify the process of finding the zeros of a polynomial.<br/> **Synthetic Division: A Powerful Tool for Polynomial Division - Q&A**$

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for finding the zeros of a polynomial and can be used to simplify the process of polynomial division. In this article, we will answer some frequently asked questions about synthetic division.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor of the form ${$ (x - c) $}$. It is a shortcut method that eliminates the need for long division and can be used to find the zeros of a polynomial.

Q: How do I use synthetic division?

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

Write down the coefficients of the polynomial in a row.
Write down the value of ${$ c $}$ in the middle of the row.
Multiply the value of ${$ c $}$ by the first coefficient and write the result below the row.
Add the result to the second coefficient and write the result below the row.
Multiply the value of ${$ c $}$ by the result and write the result below the row.
Add the result to the third coefficient and write the result below the row.
Repeat steps 5 and 6 until all coefficients have been used.
The final result is the remainder.

Q: What is the remainder in synthetic division?

A: The remainder in synthetic division is the final result after all coefficients have been used. If the remainder is zero, then the linear factor ${$ (x - c) $}$ is a factor of the polynomial.

Q: How do I determine if a linear factor is a factor of a polynomial?

A: To determine if a linear factor is a factor of a polynomial, you need to use synthetic division. If the remainder is zero, then the linear factor is a factor of the polynomial.

Q: Can I use synthetic division to find the zeros of a polynomial?

A: Yes, you can use synthetic division to find the zeros of a polynomial. If the remainder is zero, then the linear factor ${$ (x - c) $}$ is a factor of the polynomial, and ${$ c $}$ is a zero of the polynomial.

Q: How do I use synthetic division to find the zeros of a polynomial?

A: To use synthetic division to find the zeros of a polynomial, you need to follow these steps:

Write down the coefficients of the polynomial in a row.
Write down the value of ${$ c $}$ in the middle of the row.
Multiply the value of ${$ c $}$ by the first coefficient and write the result below the row.
Add the result to the second coefficient and write the result below the row.
Multiply the value of ${$ c $}$ by the result and write the result below the row.
Add the result to the third coefficient and write the result below the row.
Repeat steps 5 and 6 until all coefficients have been used.
If the remainder is zero, then ${$ c $}$ is a zero of the polynomial.

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 ${$ c $}$ in the middle of the row.
Not multiplying the value of ${$ c $}$ by the first coefficient.
Not adding the result to the second coefficient.
Not repeating steps 5 and 6 until all coefficients have been used.

Conclusion

Synthetic division is a powerful tool for polynomial division and can be used to simplify the process of finding the zeros of a polynomial. By following the steps outlined in this article, you can use synthetic division to determine if a linear factor is a factor of a polynomial and to find the zeros of a polynomial.