Select The Correct Answer.Using Synthetic Division, Find 2 X 4 + 4 X 3 + 2 X 2 + 8 X + 8 X + 2 \frac{2x^4 + 4x^3 + 2x^2 + 8x + 8}{x+2} X + 2 2 X 4 + 4 X 3 + 2 X 2 + 8 X + 8 ​ .A. 2 X 3 + 2 X + 4 2x^3 + 2x + 4 2 X 3 + 2 X + 4 B. 2 X 4 + 2 X 2 + 4 X 2x^4 + 2x^2 + 4x 2 X 4 + 2 X 2 + 4 X C. 2 X 3 + 2 X + 4 + 1 X + 2 2x^3 + 2x + 4 + \frac{1}{x+2} 2 X 3 + 2 X + 4 + X + 2 1 ​ D. 2 X 3 + 2 X 2 + 4 2x^3 + 2x^2 + 4 2 X 3 + 2 X 2 + 4

Introduction

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. In this article, we will use synthetic division to solve the problem of finding the quotient of the polynomial $2x^4 + 4x^3 + 2x^2 + 8x + 8$ divided by $x+2$.

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 often used in algebra and calculus. The method involves using a single number, called the divisor, to divide the polynomial. The divisor is usually a linear factor, such as $x+2$.

How to Perform Synthetic Division

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

1. Write down the coefficients of the polynomial: The coefficients are the numbers that multiply the powers of $x$. In this case, the coefficients are $2, 4, 2, 8, 8$.
2. Write down the divisor: The divisor is the linear factor that we are dividing by. In this case, the divisor is $x+2$.
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 $2$.
4. Multiply the divisor by the first coefficient: Multiply the divisor by the first coefficient. In this case, we multiply $x+2$ by $2$ to get $2x+4$.
5. Add the result to the next coefficient: Add the result of the multiplication to the next coefficient. In this case, we add $2x+4$ to $4$ to get $2x+6$.
6. Repeat steps 4 and 5: Repeat steps 4 and 5 until we have used up all the coefficients.
7. Write down the final result: The final result is the quotient of the polynomial divided by the divisor.

Solving the Problem

Now that we have learned how to perform synthetic division, let's use it to solve the problem of finding the quotient of the polynomial $2x^4 + 4x^3 + 2x^2 + 8x + 8$ divided by $x+2$.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are $2, 4, 2, 8, 8$.

Step 2: Write down the divisor

The divisor is $x+2$.

Step 3: Bring down the first coefficient

The first coefficient is $2$.

Step 4: Multiply the divisor by the first coefficient

We multiply $x+2$ by $2$ to get $2x+4$.

Step 5: Add the result to the next coefficient

We add $2x+4$ to $4$ to get $2x+6$.

Step 6: Repeat steps 4 and 5

We repeat steps 4 and 5 until we have used up all the coefficients.

	2	4	2	8	8
x+2	2	6	14	34	70
	2x+4	2x+6	2x+8	2x+10	2x+12

Step 7: Write down the final result

The final result is the quotient of the polynomial divided by the divisor. We can see that the quotient is $2x^3 + 2x + 4$.

Conclusion

In this article, we used synthetic division to solve the problem of finding the quotient of the polynomial $2x^4 + 4x^3 + 2x^2 + 8x + 8$ divided by $x+2$. We learned how to perform synthetic division and used it to find the quotient of the polynomial. The final result is $2x^3 + 2x + 4$.

Answer

The correct answer is:

• A. $2x^3 + 2x + 4$

Discussion

Synthetic division is a powerful tool for solving polynomial equations. It is a shortcut to the long division method and is often used in algebra and calculus. In this article, we used synthetic division to solve the problem of finding the quotient of the polynomial $2x^4 + 4x^3 + 2x^2 + 8x + 8$ divided by $x+2$. We learned how to perform synthetic division and used it to find the quotient of the polynomial. The final result is $2x^3 + 2x + 4$.

Related Topics

• Long Division: Long division is a method of dividing polynomials by linear factors. It is a more detailed method than synthetic division and is often used in algebra and calculus.
• Polynomial Equations: Polynomial equations are equations that involve polynomials. They are often used in algebra and calculus to solve problems.
• Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It is often used in science and engineering to solve problems.

References

• "Synthetic Division" by Math Open Reference. Retrieved from https://www.mathopenref.com/syntheticdivision.html
• "Polynomial Division" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4d7/x2f6f4d7
• "Algebra" by Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Algebra
  

Introduction

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. In this article, we will answer some 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 shortcut to the long division method and is often used in algebra and calculus.

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: The coefficients are the numbers that multiply the powers of $x$.
2. Write down the divisor: The divisor is the linear factor that you are dividing by.
3. Bring down the first coefficient: The first coefficient is the first number in the list of coefficients.
4. Multiply the divisor by the first coefficient: Multiply the divisor by the first coefficient.
5. Add the result to the next coefficient: Add the result of the multiplication to the next coefficient.
6. Repeat steps 4 and 5: Repeat steps 4 and 5 until you have used up all the coefficients.
7. Write down the final result: The final result is the quotient of the polynomial divided by the divisor.

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

A: Synthetic division is a shortcut to the long division method. It is faster and easier to use, but it only works for dividing polynomials by linear factors.

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

A: No, synthetic division only works for dividing polynomials by linear factors. If you need to divide a polynomial by a quadratic factor, you will need to use the long division method.

Q: How do I know if I have used up all the coefficients?

A: You will know if you have used up all the coefficients when you have added the result of the multiplication to the last coefficient.

Q: What is the final result of synthetic division?

A: The final result of synthetic division is the quotient of the polynomial divided by the divisor.

Q: Can I use synthetic division to solve polynomial equations?

A: Yes, synthetic division can be used to solve polynomial equations. It is a powerful tool for solving polynomial equations and is often used in algebra and calculus.

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: Make sure to write down the coefficients of the polynomial before starting the division.
• Not writing down the divisor: Make sure to write down the divisor before starting the division.
• Not bringing down the first coefficient: Make sure to bring down the first coefficient before starting the division.
• Not multiplying the divisor by the first coefficient: Make sure to multiply the divisor by the first coefficient before starting the division.
• Not adding the result to the next coefficient: Make sure to add the result of the multiplication to the next coefficient before starting the division.

Conclusion

Synthetic division is a powerful tool for solving polynomial equations. It is a shortcut to the long division method and is often used in algebra and calculus. In this article, we have answered some frequently asked questions about synthetic division and provided some tips for avoiding common mistakes.

Related Topics

• Long Division: Long division is a method of dividing polynomials by linear factors. It is a more detailed method than synthetic division and is often used in algebra and calculus.
• Polynomial Equations: Polynomial equations are equations that involve polynomials. They are often used in algebra and calculus to solve problems.
• Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It is often used in science and engineering to solve problems.

References

• "Synthetic Division" by Math Open Reference. Retrieved from https://www.mathopenref.com/syntheticdivision.html
• "Polynomial Division" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4d7/x2f6f4d7
• "Algebra" by Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Algebra