Complete The Division Problem Using Synthetic Division: ( − 5 X 3 − 8 X 2 + 8 ) ÷ ( X + 2 (-5x^3 - 8x^2 + 8) \div (x + 2 ( − 5 X 3 − 8 X 2 + 8 ) ÷ ( X + 2 ]

Mar 13, 2025 by ADMIN 157 views

**Complete the Division Problem Using Synthetic Division**

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 learn how to complete the division problem using synthetic division.

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 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:

Write down the coefficients of the polynomial: The coefficients of the polynomial are the numbers that multiply the variable. For example, in the polynomial $-5x^3 - 8x^2 + 8$ , the coefficients are $-5$ , $-8$ , and $8$ .
Write down the root of the linear factor: The root of the linear factor is the number that the polynomial is being divided by. In this case, the root is $-2$ .
Bring down the first coefficient: The first coefficient is the first number in the row of coefficients. In this case, the first coefficient is $-5$ .
Multiply the root by the first coefficient: Multiply the root by the first coefficient to get the next number in the row.
Add the next coefficient: Add the next coefficient to the result of the multiplication.
Repeat steps 4 and 5: Repeat steps 4 and 5 until all the coefficients have been used.
Write down the final result: The final result is the last number in the row.

Example

Let's use the polynomial $-5x^3 - 8x^2 + 8$ and divide it by $x + 2$ . We will use synthetic division to perform the division.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are $-5$ , $-8$ , and $8$ .

Step 2: Write down the root of the linear factor

The root of the linear factor is $-2$ .

Step 3: Bring down the first coefficient

The first coefficient is $-5$ .

Step 4: Multiply the root by the first coefficient

Multiply the root by the first coefficient to get $-10$ .

Step 5: Add the next coefficient

Add the next coefficient to the result of the multiplication to get $-10 + 8 = -2$ .

Step 6: Repeat steps 4 and 5

Repeat steps 4 and 5 until all the coefficients have been used.

Step 7: Write down the final result

The final result is $-2$ .

Conclusion

In this article, we learned how to complete the division problem using synthetic division. We used the polynomial $-5x^3 - 8x^2 + 8$ and divided it by $x + 2$ . We used synthetic division to perform the division and obtained a final result of $-2$ .

Synthetic Division Formula

The synthetic division formula is:

\begin{array}{c|cccc} c_0 & c_1 & c_2 & \cdots & c_n \\ & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \\ \hline c_0 & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \end{array}

where $c_i$ are the coefficients of the polynomial and $c_0$ is the root of the linear factor.

Synthetic Division Example

Let's use the polynomial $-5x^3 - 8x^2 + 8$ and divide it by $x + 2$ . We will use synthetic division to perform the division.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are $-5$ , $-8$ , and $8$ .

Step 2: Write down the root of the linear factor

The root of the linear factor is $-2$ .

Step 3: Bring down the first coefficient

The first coefficient is $-5$ .

Step 4: Multiply the root by the first coefficient

Multiply the root by the first coefficient to get $-10$ .

Step 5: Add the next coefficient

Add the next coefficient to the result of the multiplication to get $-10 + 8 = -2$ .

Step 6: Repeat steps 4 and 5

Repeat steps 4 and 5 until all the coefficients have been used.

Step 7: Write down the final result

The final result is $-2$ .

Synthetic Division Table

Coefficient	Root	Result
$-5$	$-2$	$-10$
$-8$	$-2$	$-2$
$8$	$-2$	$0$

Synthetic Division Conclusion

Synthetic Division Formula

The synthetic division formula is:

\begin{array}{c|cccc} c_0 & c_1 & c_2 & \cdots & c_n \\ & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \\ \hline c_0 & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \end{array}

where $c_i$ are the coefficients of the polynomial and $c_0$ is the root of the linear factor.

Synthetic Division Example

Let's use the polynomial $-5x^3 - 8x^2 + 8$ and divide it by $x + 2$ . We will use synthetic division to perform the division.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are $-5$ , $-8$ , and $8$ .

Step 2: Write down the root of the linear factor

The root of the linear factor is $-2$ .

Step 3: Bring down the first coefficient

The first coefficient is $-5$ .

Step 4: Multiply the root by the first coefficient

Multiply the root by the first coefficient to get $-10$ .

Step 5: Add the next coefficient

Add the next coefficient to the result of the multiplication to get $-10 + 8 = -2$ .

Step 6: Repeat steps 4 and 5

Repeat steps 4 and 5 until all the coefficients have been used.

Step 7: Write down the final result

The final result is $-2$ .

Synthetic Division Practice

Practice synthetic division by using the following polynomials and dividing them by the given linear factors.

$2x^3 + 3x^2 - 4x + 1$ divided by $x + 1$
$x^3 - 2x^2 + 3x - 4$ divided by $x - 1$
$x^3 + 2x^2 - 3x + 1$ divided by $x + 1$

Synthetic Division Conclusion

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:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Bring down the first coefficient.
Multiply the root by the first coefficient.
Add the next coefficient.
Repeat steps 4 and 5 until all the coefficients have been used.
Write down the final result.

Q: What is the synthetic division formula?

A: The synthetic division formula is:

\begin{array}{c|cccc} c_0 & c_1 & c_2 & \cdots & c_n \\ & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \\ \hline c_0 & c_0c_1 & c_0c_2 + c_1 & \cdots & c_0c_n + c_{n-1} \end{array}

where $c_i$ are the coefficients of the polynomial and $c_0$ is the root of the linear factor.

Q: How do I use synthetic division to divide a polynomial by a linear factor?

A: To use synthetic division to divide a polynomial by a linear factor, you need to follow these steps:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Bring down the first coefficient.
Multiply the root by the first coefficient.
Add the next coefficient.
Repeat steps 4 and 5 until all the coefficients have been used.
Write down the final result.

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

A: Synthetic division is a shortcut to the long division method. It is used to divide polynomials by linear factors, while long division is used to divide polynomials by any factor.

Q: Can I use synthetic division to divide a polynomial by a quadratic factor?

A: No, synthetic division is only used to divide polynomials by linear factors. If you need to divide a polynomial by a quadratic factor, you will need to use a different method, such as factoring or the quadratic formula.

Q: How do I know if a polynomial can be divided by a linear factor using synthetic division?

A: A polynomial can be divided by a linear factor using synthetic division if the linear factor is a factor of the polynomial. You can check if a linear factor is a factor of a polynomial by using the remainder theorem.

Q: What is the remainder theorem?

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

Q: How do I use the remainder theorem to check if a linear factor is a factor of a polynomial?

A: To use the remainder theorem to check if a linear factor is a factor of a polynomial, you need to follow these steps:

Write down the polynomial.
Write down the linear factor.
Substitute the root of the linear factor into the polynomial.
Evaluate the polynomial at the root.
If the result is equal to zero, then the linear factor is a factor of the polynomial.

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

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

Not bringing down the first coefficient.
Not multiplying the root by the first coefficient.
Not adding the next coefficient.
Not repeating steps 4 and 5 until all the coefficients have been used.
Not writing down the final result.

Q: How do I practice synthetic division?

A: You can practice synthetic division by using the following polynomials and dividing them by the given linear factors:

$2x^3 + 3x^2 - 4x + 1$ divided by $x + 1$
$x^3 - 2x^2 + 3x - 4$ divided by $x - 1$
$x^3 + 2x^2 - 3x + 1$ divided by $x + 1$

You can also use online resources or worksheets to practice synthetic division.

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

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

Dividing polynomials to find the roots of a polynomial equation.
Finding the maximum or minimum value of a polynomial function.
Dividing polynomials to find the equation of a tangent line to a curve.
Dividing polynomials to find the equation of a normal line to a curve.

Q: Can I use synthetic division to divide a polynomial by a rational factor?

A: No, synthetic division is only used to divide polynomials by linear factors. If you need to divide a polynomial by a rational factor, you will need to use a different method, such as factoring or the rational root theorem.

Q: What is the rational root theorem?

A: The rational root theorem is a theorem that states that if a polynomial $f(x)$ has integer coefficients, then any rational root of $f(x)$ must be of the form $p/q$ , where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Q: How do I use the rational root theorem to find the rational roots of a polynomial?

A: To use the rational root theorem to find the rational roots of a polynomial, you need to follow these steps:

Write down the polynomial.
Write down the factors of the constant term.
Write down the factors of the leading coefficient.
List all possible rational roots.
Test each possible rational root by substituting it into the polynomial.
If the result is equal to zero, then the rational root is a root of the polynomial.

Q: What are some common mistakes to avoid when using the rational root theorem?

A: Some common mistakes to avoid when using the rational root theorem include:

Not listing all possible rational roots.
Not testing each possible rational root.
Not substituting the rational root into the polynomial.
Not checking if the result is equal to zero.

Q: How do I practice the rational root theorem?

A: You can practice the rational root theorem by using the following polynomials and finding their rational roots:

$x^3 + 2x^2 - 3x + 1$ divided by $x + 1$
$x^3 - 2x^2 + 3x - 4$ divided by $x - 1$
$x^3 + 2x^2 - 3x + 1$ divided by $x + 1$

You can also use online resources or worksheets to practice the rational root theorem.