Divide:$ \left(9 - 17x + 15x^3\right) \div \left(3x^2 - 4\right) }$Write Your Answer In The Following Form Quotient { + \frac{\text{Remainder }{3x^2 - 4} $}$.

$$

Introduction

Polynomial Division is a fundamental concept in algebra, and it plays a crucial role in solving equations and manipulating expressions. In this article, we will focus on dividing a polynomial by another polynomial, specifically $\left(9 - 17x + 15x^3\right)$ divided by $\left(3x^2 - 4\right)$. We will use the long division method to find the quotient and remainder.

Long Division Method

The long division method is a step-by-step process for dividing polynomials. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Step 1: Divide the Highest Degree Term

To start the long division, we divide the highest degree term of the dividend, $15x^3$, by the highest degree term of the divisor, $3x^2$. This gives us $5x$.

Step 2: Multiply the Divisor by the Result

Next, we multiply the entire divisor, $3x^2 - 4$, by the result, $5x$. This gives us $15x^3 - 20x$.

Step 3: Subtract the Result from the Dividend

We subtract the result from the dividend, $9 - 17x + 15x^3 - 20x$. This gives us $9 - 17x - 20x + 15x^3 - 15x^3$, which simplifies to $9 - 37x$.

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend, $9$, by the highest degree term of the divisor, $3x^2$. This gives us $0$.

Step 5: Write the Quotient and Remainder

Since the degree of the remainder, $9 - 37x$, is less than the degree of the divisor, $3x^2 - 4$, we can stop the long division process. The quotient is $5x$ and the remainder is $9 - 37x$.

Writing the Answer in the Required Form

We are asked to write the answer in the following form: Quotient $+ \frac{\text{Remainder}}{3x^2 - 4}$. Therefore, the final answer is:

$$5x + \frac{9 - 37x}{3x^2 - 4}$$

Conclusion

In this article, we used the long division method to divide the polynomial $\left(9 - 17x + 15x^3\right)$ by the polynomial $\left(3x^2 - 4\right)$. We found the quotient to be $5x$ and the remainder to be $9 - 37x$. We then wrote the answer in the required form, which is Quotient $+ \frac{\text{Remainder}}{3x^2 - 4}$.

Example Use Case

The long division method is a powerful tool for solving equations and manipulating expressions. For example, suppose we want to solve the equation $15x^3 - 17x + 9 = (3x^2 - 4)(5x + k)$. We can use the long division method to find the value of $k$.

Tips and Tricks

• When using the long division method, make sure to divide the highest degree term of the dividend by the highest degree term of the divisor.
• When multiplying the divisor by the result, make sure to multiply the entire divisor by the result.
• When subtracting the result from the dividend, make sure to subtract the result from the dividend.
• When repeating the process, make sure to divide the highest degree term of the new dividend by the highest degree term of the divisor.

Frequently Asked Questions

• What is the long division method? The long division method is a step-by-step process for dividing polynomials.
• How do I use the long division method? To use the long division method, divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the entire divisor by the result, subtract the result from the dividend, and repeat the process until the degree of the remainder is less than the degree of the divisor.
• What is the quotient and remainder? The quotient is the result of the division, and the remainder is the amount left over after the division.

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Long Division of Polynomials" by Purplemath
• [3] "Dividing Polynomials" by Mathway

Further Reading

• "Polynomial Division" by Khan Academy
• "Long Division of Polynomials" by IXL
• "Dividing Polynomials" by CK-12
  

Introduction

Polynomial division is a fundamental concept in algebra, and it plays a crucial role in solving equations and manipulating expressions. In this article, we will answer some of the most frequently asked questions about polynomial division.

Q: What is polynomial division?

A: Polynomial division is the process of dividing one polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Q: What is the long division method?

A: The long division method is a step-by-step process for dividing polynomials. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, subtracting the result from the dividend, and repeating the process until the degree of the remainder is less than the degree of the divisor.

Q: How do I use the long division method?

A: To use the long division method, follow these steps:

1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
2. Multiply the entire divisor by the result.
3. Subtract the result from the dividend.
4. Repeat the process until the degree of the remainder is less than the degree of the divisor.

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.

Q: How do I write the answer in the required form?

A: To write the answer in the required form, Quotient $+ \frac{\text{Remainder}}{3x^2 - 4}$, simply write the quotient and the remainder divided by the divisor.

Q: What are some common mistakes to avoid when using the long division method?

A: Some common mistakes to avoid when using the long division method include:

• Not dividing the highest degree term of the dividend by the highest degree term of the divisor.
• Not multiplying the entire divisor by the result.
• Not subtracting the result from the dividend.
• Not repeating the process until the degree of the remainder is less than the degree of the divisor.

Q: How do I check my work when using the long division method?

A: To check your work when using the long division method, multiply the quotient by the divisor and add the remainder. If the result is equal to the dividend, then your work is correct.

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

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

• Solving equations
• Manipulating expressions
• Finding the roots of a polynomial
• Factoring polynomials

Q: How do I use polynomial division to solve equations?

A: To use polynomial division to solve equations, follow these steps:

1. Divide the polynomial by the divisor.
2. Set the quotient equal to the unknown variable.
3. Solve for the unknown variable.

Q: How do I use polynomial division to manipulate expressions?

A: To use polynomial division to manipulate expressions, follow these steps:

1. Divide the polynomial by the divisor.
2. Simplify the result.

Q: What are some common mistakes to avoid when using polynomial division to solve equations?

A: Some common mistakes to avoid when using polynomial division to solve equations include:

• Not dividing the polynomial by the divisor.
• Not setting the quotient equal to the unknown variable.
• Not solving for the unknown variable.

Q: What are some common mistakes to avoid when using polynomial division to manipulate expressions?

A: Some common mistakes to avoid when using polynomial division to manipulate expressions include:

• Not dividing the polynomial by the divisor.
• Not simplifying the result.

Q: How do I check my work when using polynomial division to solve equations?

A: To check your work when using polynomial division to solve equations, follow these steps:

1. Multiply the quotient by the divisor.
2. Add the remainder.
3. If the result is equal to the original equation, then your work is correct.

Q: How do I check my work when using polynomial division to manipulate expressions?

A: To check your work when using polynomial division to manipulate expressions, follow these steps:

1. Simplify the result.
2. If the result is equal to the original expression, then your work is correct.

Conclusion

In this article, we have answered some of the most frequently asked questions about polynomial division. We have covered topics such as the long division method, the quotient and remainder, and common mistakes to avoid. We have also provided examples of real-world applications of polynomial division and tips for checking your work.