Use Synthetic Division To Solve ( 4 X 3 − 3 X 2 + 5 X + 6 ) ÷ ( X + 6 \left(4 X^3-3 X^2+5 X+6\right) \div(x+6 ( 4 X 3 − 3 X 2 + 5 X + 6 ) ÷ ( X + 6 ]. What Is The Quotient?A. 4 X 2 − 27 X + 167 − 996 X − 6 4 X^2-27 X+167-\frac{996}{x-6} 4 X 2 − 27 X + 167 − X − 6 996 ​ B. 4 X 2 + 21 X + 131 + 792 X + 6 4 X^2+21 X+131+\frac{792}{x+6} 4 X 2 + 21 X + 131 + X + 6 792 ​ C. 4 X 2 + 21 X + 131 + 792 X − 6 4 X^2+21 X+131+\frac{792}{x-6} 4 X 2 + 21 X + 131 + X − 6 792 ​ D. $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 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 problem of dividing the polynomial $\left(4 x^3-3 x^2+5 x+6\right)$ by $(x+6)$ 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, -6) below the row of coefficients.
3. Bring down the first coefficient (in this case, 4).
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 you reach the last coefficient.
7. The final result is the quotient, with the remainder written below the last coefficient.

Solving the Problem

Now that we have learned how to perform synthetic division, let's use it to solve the problem of dividing the polynomial $\left(4 x^3-3 x^2+5 x+6\right)$ by $(x+6)$.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are 4, -3, 5, and 6.

Step 2: Write down the value of the linear factor

The value of the linear factor is -6.

Step 3: Bring down the first coefficient

The first coefficient is 4.

Step 4: Multiply the value of the linear factor by the first coefficient

The result of multiplying -6 by 4 is -24.

Step 5: Add the second coefficient and the result from step 4

The second coefficient is -3. Adding -3 and -24 gives -27.

Step 6: Repeat steps 4 and 5 for each coefficient

The results of the next steps are:

• Multiply -6 by -27: 162
• Add 5 and 162: 167
• Multiply -6 by 167: -1002
• Add 6 and -1002: -996

Step 7: Write down the quotient and remainder

The quotient is $4 x^2-27 x+167$ and the remainder is $\frac{-996}{x+6}$.

Conclusion

In this article, we used synthetic division to solve the problem of dividing the polynomial $\left(4 x^3-3 x^2+5 x+6\right)$ by $(x+6)$ and found the quotient to be $4 x^2-27 x+167-\frac{996}{x+6}$. This is the correct answer, which can be verified by using the long division method.

Answer

The correct answer is:

$$

Introduction

In our previous article, we used synthetic division to solve the problem of dividing the polynomial $\left(4 x^3-3 x^2+5 x+6\right)$ by $(x+6)$ and found the quotient to be $4 x^2-27 x+167-\frac{996}{x+6}$. In this article, we will answer some frequently asked questions about synthetic division and provide additional examples to help you understand the concept better.

Q&A

Q: What is synthetic division used for?

A: Synthetic division is 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).

Q: How do I know if I should use synthetic division or long division?

A: If the divisor is a linear factor of the form (x - a) or (x + a), you should use synthetic division. If the divisor is not a linear factor, you should use long division.

Q: What are the steps involved in synthetic division?

A: The steps involved in synthetic division are:

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, -6) below the row of coefficients.
3. Bring down the first coefficient (in this case, 4).
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 you reach the last coefficient.
7. The final result is the quotient, with the remainder written below the last coefficient.

Q: What is the remainder in synthetic division?

A: The remainder in synthetic division is the value that is left over after the division. It is written below the last coefficient.

Q: How do I know if the remainder is zero or not?

A: If the remainder is zero, it means that the polynomial is divisible by the linear factor. If the remainder is not zero, it means that the polynomial is not divisible by the linear factor.

Q: Can I use synthetic division to divide polynomials by factors of the form (x^2 + ax + b)?

A: No, you cannot use synthetic division to divide polynomials by factors of the form (x^2 + ax + b). Synthetic division is only used to divide polynomials by linear factors.

Q: Can I use synthetic division to divide polynomials by factors of the form (x - a)^n?

A: No, you cannot use synthetic division to divide polynomials by factors of the form (x - a)^n. Synthetic division is only used to divide polynomials by linear factors.

Examples

Example 1: Divide the polynomial $\left(x^3-2 x^2+3 x-4\right)$ by $(x-2)$

Using synthetic division, we get:

• Multiply -2 by 1: -2
• Add -2 and -2: -4
• Multiply -2 by -4: 8
• Add 3 and 8: 11
• Multiply -2 by 11: -22
• Add -4 and -22: -26

The quotient is $x^2+11 x+11$ and the remainder is $\frac{-26}{x-2}$.

Example 2: Divide the polynomial $\left(2 x^3+3 x^2-4 x+5\right)$ by $(x+2)$

Using synthetic division, we get:

• Multiply -2 by 2: -4
• Add 3 and -4: -1
• Multiply -2 by -1: 2
• Add -4 and 2: -2
• Multiply -2 by -2: 4
• Add 5 and 4: 9

The quotient is $2 x^2- x-2$ and the remainder is $\frac{9}{x+2}$.

Conclusion

In this article, we answered some frequently asked questions about synthetic division and provided additional examples to help you understand the concept better. Synthetic division is a powerful tool for polynomial division and is particularly useful when dividing polynomials by linear factors. We hope that this article has helped you to understand synthetic division better and to use it to solve problems in mathematics.