What Is The Remainder When 2 Is Synthetically Divided Into The Polynomial $-3x^2 + 7x - 9$?A. -5 B. 0 C. -7 D. 3

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Introduction

Synthetic division is a method used to divide a polynomial by a linear factor of the form (x - a). It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we will explore the concept of synthetic division and use it to find the remainder when 2 is synthetically divided into the polynomial $-3x^2 + 7x - 9$.

What is Synthetic Division?

Synthetic division is a method of dividing a polynomial by a linear factor of the form (x - a). It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). The process of synthetic division involves writing the coefficients of the polynomial in a row, followed by the value of a, and then performing a series of multiplications and additions to find the remainder.

How to Perform Synthetic Division

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

1. Write the coefficients of the polynomial in a row.
2. Write the value of a in the next column.
3. Multiply the value of a by the first coefficient and write the result below the second coefficient.
4. Add the second coefficient and the result from step 3.
5. Multiply the value of a by the result from step 4 and write the result below the third coefficient.
6. Add the third coefficient and the result from step 5.
7. Repeat steps 5 and 6 until we have added all the coefficients.
8. The final result is the remainder.

Example: Synthetic Division of $-3x^2 + 7x - 9$ by (x - 2)

Let's use the polynomial $-3x^2 + 7x - 9$ and divide it by (x - 2) using synthetic division.

-3	7	-9
2
-6
	15
		-15

Step 1: Multiply the value of a by the first coefficient

The value of a is 2, and the first coefficient is -3. Multiply 2 by -3 to get -6.

Step 2: Add the second coefficient and the result from step 1

The second coefficient is 7, and the result from step 1 is -6. Add 7 and -6 to get 1.

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

The value of a is 2, and the result from step 2 is 1. Multiply 2 by 1 to get 2.

Step 4: Add the third coefficient and the result from step 3

The third coefficient is -9, and the result from step 3 is 2. Add -9 and 2 to get -7.

Conclusion

The remainder when 2 is synthetically divided into the polynomial $-3x^2 + 7x - 9$ is -7.

Final Answer

The final answer is C. -7.

Discussion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we used synthetic division to find the remainder when 2 is synthetically divided into the polynomial $-3x^2 + 7x - 9$. The remainder is -7.

Related Topics

• Synthetic division
• Polynomial division
• Linear factors
• Remainder theorem

References

• [1] "Synthetic Division" by Math Open Reference
• [2] "Polynomial Division" by Khan Academy
• [3] "Linear Factors" by Wolfram MathWorld

Keywords

• Synthetic division
• Polynomial division
• Linear factors
• Remainder theorem
• Math
• Algebra
• Polynomials
• Division
• Remainder
  

Introduction

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (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 a polynomial by a linear factor of the form (x - a). It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a).

Q: How do I perform synthetic division?

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

1. Write the coefficients of the polynomial in a row.
2. Write the value of a in the next column.
3. Multiply the value of a by the first coefficient and write the result below the second coefficient.
4. Add the second coefficient and the result from step 3.
5. Multiply the value of a by the result from step 4 and write the result below the third coefficient.
6. Add the third coefficient and the result from step 5.
7. Repeat steps 5 and 6 until you have added all the coefficients.
8. The final result is the remainder.

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a).

Q: How do I use synthetic division to find the remainder?

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

1. Write the coefficients of the polynomial in a row.
2. Write the value of a in the next column.
3. Multiply the value of a by the first coefficient and write the result below the second coefficient.
4. Add the second coefficient and the result from step 3.
5. Multiply the value of a by the result from step 4 and write the result below the third coefficient.
6. Add the third coefficient and the result from step 5.
7. Repeat steps 5 and 6 until you have added all the coefficients.
8. The final result is the remainder.

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

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

• Not writing the coefficients of the polynomial in the correct order.
• Not writing the value of a in the correct column.
• Not multiplying the value of a by the correct coefficient.
• Not adding the correct coefficients.
• Not repeating the process until you have added all the coefficients.

Q: How do I check my work when performing synthetic division?

A: To check your work when performing synthetic division, you can use the following steps:

1. Write the coefficients of the polynomial in a row.
2. Write the value of a in the next column.
3. Multiply the value of a by the first coefficient and write the result below the second coefficient.
4. Add the second coefficient and the result from step 3.
5. Multiply the value of a by the result from step 4 and write the result below the third coefficient.
6. Add the third coefficient and the result from step 5.
7. Repeat steps 5 and 6 until you have added all the coefficients.
8. The final result is the remainder.

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

A: Synthetic division has many real-world applications, including:

• Finding the remainder when a polynomial is divided by a linear factor.
• Finding the roots of a polynomial.
• Finding the maximum or minimum value of a polynomial.
• Finding the rate of change of a polynomial.

Conclusion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we have answered some frequently asked questions about synthetic division.

Final Answer

The final answer is that synthetic division is a useful tool for dividing polynomials by linear factors.

Discussion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we have answered some frequently asked questions about synthetic division.

Related Topics

• Synthetic division
• Polynomial division
• Linear factors
• Remainder theorem
• Math
• Algebra
• Polynomials
• Division
• Remainder

References

• [1] "Synthetic Division" by Math Open Reference
• [2] "Polynomial Division" by Khan Academy
• [3] "Linear Factors" by Wolfram MathWorld

Keywords

• Synthetic division
• Polynomial division
• Linear factors
• Remainder theorem
• Math
• Algebra
• Polynomials
• Division
• Remainder