Use Synthetic Division To Solve \[$(3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2)\$\]. What Is The Quotient?A. \[$3x^3 + 12x^2 + 26x + 61 + \frac{132}{x+2}\$\]B. \[$3x^3 + 2x + 5\$\]C. \[$3x^3 + 12x^2 + 26x + 61 +

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Introduction to Synthetic Division

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) or (x + a). In this article, we will use synthetic division to solve the polynomial equation (3x4+6x3+2x2+9x+10)÷(x+2)(3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2) 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) or (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 linear factor (in this case, x + 2) on the left.
  3. Bring down the first coefficient (in this case, 3).
  4. Multiply the value of the linear 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. Repeat steps 4 and 5 for each coefficient, until we reach the last coefficient.
  7. The final result is the quotient, with the remainder written below.

Solving the Polynomial Equation

Now, let's use synthetic division to solve the polynomial equation (3x4+6x3+2x2+9x+10)÷(x+2)(3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2).

Step 1: Write Down the Coefficients

The coefficients of the polynomial are 3, 6, 2, 9, and 10. We write these down in a row, with the constant term on the right.

| 3 | 6 | 2 | 9 | 10 |

Step 2: Write Down the Value of the Linear Factor

The value of the linear factor is x + 2. We write this down on the left.

x + 2

Step 3: Bring Down the First Coefficient

We bring down the first coefficient, which is 3.

3 6 2 9 10
3

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

We multiply the value of the linear factor (x + 2) by the first coefficient (3). This gives us 3x + 6. We write this below the second coefficient.

3 6 2 9 10
3 3x + 6

Step 5: Add the Second Coefficient and the Result from Step 4

We add the second coefficient (6) and the result from step 4 (3x + 6). This gives us 3x + 12. We write this below the third coefficient.

3 6 2 9 10
3 3x + 6 3x + 12

Step 6: Repeat Steps 4 and 5 for Each Coefficient

We repeat steps 4 and 5 for each coefficient, until we reach the last coefficient.

3 6 2 9 10
3 3x + 6 3x + 12 3x + 18 3x + 24

Step 7: Write Down the Quotient and Remainder

The final result is the quotient, with the remainder written below. The quotient is 3x^3 + 12x^2 + 26x + 61, and the remainder is 132/(x + 2).

Conclusion

In this article, we used synthetic division to solve the polynomial equation (3x4+6x3+2x2+9x+10)÷(x+2)(3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2) and found the quotient. The quotient is 3x^3 + 12x^2 + 26x + 61, and the remainder is 132/(x + 2). This is a powerful tool for solving polynomial equations, and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a).

Final Answer

The final answer is: A. 3x^3 + 12x^2 + 26x + 61 + \frac{132}{x+2}

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) or (x + a). 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 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) or (x + a).

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 in a row, with the constant term on the right.
  2. Write down the value of the linear factor (in this case, x + 2) on the left.
  3. Bring down the first coefficient (in this case, 3).
  4. Multiply the value of the linear 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. Repeat steps 4 and 5 for each coefficient, until we reach the last coefficient.
  7. The final result is the quotient, with the remainder written below.

Q: What is the Quotient and Remainder?

A: The quotient is the result of the division, and the remainder is the amount left over after the division. In the example we used earlier, the quotient is 3x^3 + 12x^2 + 26x + 61, and the remainder is 132/(x + 2).

Q: When Should I Use Synthetic Division?

A: You should use synthetic division when dividing polynomials by factors of the form (x - a) or (x + a). It is particularly useful when the divisor is a linear factor, as it can save time and effort compared to the long division method.

Q: Can I Use Synthetic Division with Other Types of Divisors?

A: No, synthetic division is only used with linear factors of the form (x - a) or (x + a). If you need to divide a polynomial by a quadratic or higher-degree factor, you will need to use a different method, such as long division or factoring.

Q: How Do I Check My Work?

A: To check your work, you can multiply the quotient by the divisor and add the remainder. If the result is the original polynomial, then your work is correct.

Q: What are Some Common Mistakes to Avoid?

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

  • Forgetting to bring down the first coefficient
  • Not multiplying the value of the linear factor by the first coefficient
  • Not adding the second coefficient and the result from step 4
  • Not repeating steps 4 and 5 for each coefficient
  • Not writing down the remainder correctly

Conclusion

Synthetic division is a powerful tool for solving polynomial equations, and is particularly useful when dividing polynomials by factors of the form (x - a) or (x + a). By following the steps outlined in this article, you can use synthetic division to solve a wide range of polynomial equations.

Final Tips

  • Practice, practice, practice! The more you practice using synthetic division, the more comfortable you will become with the method.
  • Make sure to check your work carefully to avoid mistakes.
  • Don't be afraid to ask for help if you are struggling with synthetic division.

Common Synthetic Division Problems

Here are some common synthetic division problems to try:

  • (x3+2x2+3x+4)÷(x+1)(x^3 + 2x^2 + 3x + 4) \div (x + 1)
  • (x4+3x3+2x2+x+1)÷(x+1)(x^4 + 3x^3 + 2x^2 + x + 1) \div (x + 1)
  • (x5+2x4+3x3+4x2+5x+6)÷(x+1)(x^5 + 2x^4 + 3x^3 + 4x^2 + 5x + 6) \div (x + 1)

Try solving these problems using synthetic division, and then check your work to make sure you are correct.