Use Synthetic Division To Decide Whether The Given Number Is A Zero Of The Polynomial Function.Given Number: 2 7 \frac{2}{7} 7 2 ​ Polynomial Function: F ( X ) = 7 X 4 + 3 X 3 − X + 1 F(x) = 7x^4 + 3x^3 - X + 1 F ( X ) = 7 X 4 + 3 X 3 − X + 1

Introduction

In algebra, synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for finding the zeros of a polynomial function. In this article, we will use synthetic division to determine whether a given number is a zero of a polynomial function. The given number is $\frac{2}{7}$, and the polynomial function is $f(x) = 7x^4 + 3x^3 - x + 1$.

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. The process involves writing down the coefficients of the polynomial, followed by the value of $c$, and then performing a series of multiplications and additions.

How to Perform Synthetic Division

To perform synthetic division, follow these steps:

1. Write down the coefficients of the polynomial, starting with the coefficient of the highest degree term.
2. Write down the value of $c$.
3. Multiply the value of $c$ by the first coefficient, and write the result below the second coefficient.
4. Add the two coefficients together, and write the result below the second coefficient.
5. Multiply the value of $c$ by the result from step 4, and write the result below the third coefficient.
6. Add the two coefficients together, and write the result below the third coefficient.
7. Repeat steps 5 and 6 until you reach the last coefficient.
8. The final result is the remainder of the division.

Performing Synthetic Division on the Given Polynomial

Let's perform synthetic division on the given polynomial $f(x) = 7x^4 + 3x^3 - x + 1$ using the value of $c = \frac{2}{7}$.

	7	3	-1	1
$\frac{2}{7}$	$\frac{14}{7}$	$\frac{6}{7}$	$\frac{-2}{7}$	$\frac{2}{7}$

Step 1: Multiply the value of $c$ by the first coefficient

$\frac{2}{7} \times 7 = 2$

Step 2: Add the two coefficients together

$2 + 3 = 5$

Step 3: Multiply the value of $c$ by the result from step 2

$\frac{2}{7} \times 5 = \frac{10}{7}$

Step 4: Add the two coefficients together

$\frac{10}{7} + 3 = \frac{29}{7}$

Step 5: Multiply the value of $c$ by the result from step 4

$\frac{2}{7} \times \frac{29}{7} = \frac{58}{49}$

Step 6: Add the two coefficients together

$\frac{58}{49} + (-1) = \frac{9}{49}$

Step 7: Multiply the value of $c$ by the result from step 6

$\frac{2}{7} \times \frac{9}{49} = \frac{18}{343}$

Step 8: Add the two coefficients together

$\frac{18}{343} + 1 = \frac{361}{343}$

Conclusion

The final result of the synthetic division is $\frac{361}{343}$. This means that the given number $\frac{2}{7}$ is not a zero of the polynomial function $f(x) = 7x^4 + 3x^3 - x + 1$.

Why is Synthetic Division Important?

Synthetic division is an important tool in algebra because it allows us to quickly and easily divide polynomials by linear factors. This is particularly useful when we need to find the zeros of a polynomial function. By using synthetic division, we can avoid the tedious process of long division and get the result much faster.

Real-World Applications of Synthetic Division

Synthetic division has many real-world applications in fields such as engineering, physics, and computer science. For example, it can be used to model population growth, electrical circuits, and mechanical systems. It can also be used to solve systems of linear equations and to find the roots of quadratic equations.

Conclusion

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Q&A: 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.

Q: How do I perform synthetic division?

A: To perform synthetic division, follow these steps:

1. Write down the coefficients of the polynomial, starting with the coefficient of the highest degree term.
2. Write down the value of $c$.
3. Multiply the value of $c$ by the first coefficient, and write the result below the second coefficient.
4. Add the two coefficients together, and write the result below the second coefficient.
5. Multiply the value of $c$ by the result from step 4, and write the result below the third coefficient.
6. Add the two coefficients together, and write the result below the third coefficient.
7. Repeat steps 5 and 6 until you reach the last coefficient.
8. The final result is the remainder of the division.

Q: What is the purpose of synthetic division?

A: The purpose of synthetic division is to quickly and easily divide polynomials by linear factors. This is particularly useful when we need to find the zeros of a polynomial function.

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

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

1. Write down the polynomial function.
2. Choose a value of $c$ to test.
3. Perform synthetic division using the value of $c$.
4. If the remainder is zero, then the value of $c$ is a zero of the polynomial function.
5. If the remainder is not zero, then the value of $c$ is not a zero of the polynomial function.

Q: What are some real-world applications of synthetic division?

A: Synthetic division has many real-world applications in fields such as engineering, physics, and computer science. For example, it can be used to model population growth, electrical circuits, and mechanical systems. It can also be used to solve systems of linear equations and to find the roots of quadratic equations.

Q: Can I use synthetic division to divide polynomials by other types of factors?

A: No, synthetic division is only used to divide polynomials by linear factors of the form $(x - c)$. If you need to divide a polynomial by a different type of factor, you will need to use a different method.

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

A: To check if you have performed synthetic division correctly, follow these steps:

1. Make sure that you have written down the coefficients of the polynomial correctly.
2. Make sure that you have written down the value of $c$ correctly.
3. Make sure that you have performed the multiplications and additions correctly.
4. Make sure that the final result is the remainder of the division.

Q: What are some common mistakes to avoid when performing synthetic division?

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

• Writing down the coefficients of the polynomial incorrectly.
• Writing down the value of $c$ incorrectly.
• Performing the multiplications and additions incorrectly.
• Not checking the final result to make sure it is the remainder of the division.

Conclusion

In conclusion, synthetic division is a powerful tool for finding the zeros of a polynomial function. By using synthetic division, we can quickly and easily divide polynomials by linear factors and determine whether a given number is a zero of the polynomial function. In this article, we answered some frequently asked questions about synthetic division and provided some tips and tricks for performing synthetic division correctly.