Divide Using Long Division And State The Quotient, $q$.$\[ \frac{9x^3 - 5x^2 + 5x + 1}{9x^2 + 4x + 3} \\]$\[ \frac{9x^3 - 5x^2 + 5x + 1}{9x^2 + 4x + 3} = \square + \frac{\square}{9x^2 + 4x + 3} \\](Simplify Your Answers. Do

Introduction

In this article, we will learn how to divide a polynomial using long division. Long division is a method of dividing a polynomial by another polynomial of lower degree. It is a powerful tool in algebra that helps us simplify complex expressions and solve equations. We will use the long division method to divide the polynomial $9x^3 - 5x^2 + 5x + 1$ by $9x^2 + 4x + 3$ and state the quotient, $q$.

Step 1: Write the Dividend and Divisor

To begin the long division process, we need to write the dividend and divisor in the correct format. The dividend is the polynomial that we want to divide, and the divisor is the polynomial by which we are dividing.

${ \frac{9x^3 - 5x^2 + 5x + 1}{9x^2 + 4x + 3} }$

Step 2: Divide the Leading Term

The first step in long division is to divide the leading term of the dividend by the leading term of the divisor. In this case, the leading term of the dividend is $9x^3$, and the leading term of the divisor is $9x^2$.

${ \frac{9x^3}{9x^2} = x }$

Step 3: Multiply the Divisor by the Quotient

Next, we multiply the divisor by the quotient obtained in the previous step.

${ (9x^2 + 4x + 3) \times x = 9x^3 + 4x^2 + 3x }$

Step 4: Subtract the Product from the Dividend

We then subtract the product obtained in the previous step from the dividend.

${ (9x^3 - 5x^2 + 5x + 1) - (9x^3 + 4x^2 + 3x) = -9x^2 + 2x + 1 }$

Step 5: Bring Down the Next Term

We bring down the next term of the dividend, which is $0$ in this case.

Step 6: Repeat the Process

We repeat the process of dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by the quotient, subtracting the product from the dividend, and bringing down the next term.

${ \frac{-9x^2}{9x^2} = -1 }$

${ (-9x^2 + 4x + 3) \times (-1) = -9x^2 - 4x - 3 }$

${ (-9x^2 + 2x + 1) - (-9x^2 - 4x - 3) = 6x + 4 }$

Step 7: Write the Quotient and Remainder

We can now write the quotient and remainder.

${ \frac{9x^3 - 5x^2 + 5x + 1}{9x^2 + 4x + 3} = x - 1 + \frac{6x + 4}{9x^2 + 4x + 3} }$

Conclusion

In this article, we learned how to divide a polynomial using long division. We used the long division method to divide the polynomial $9x^3 - 5x^2 + 5x + 1$ by $9x^2 + 4x + 3$ and stated the quotient, $q$. We also learned how to write the quotient and remainder in the correct format.

Example Problems

1. Divide the polynomial $2x^3 + 5x^2 + 3x + 1$ by $x^2 + 2x + 1$.
2. Divide the polynomial $3x^3 - 2x^2 + 4x + 1$ by $x^2 + x + 1$.

Solutions

1. ${ \frac{2x^3 + 5x^2 + 3x + 1}{x^2 + 2x + 1} = 2x + 1 + \frac{3x + 1}{x^2 + 2x + 1} }$
2. ${ \frac{3x^3 - 2x^2 + 4x + 1}{x^2 + x + 1} = 3x - 1 + \frac{5x + 2}{x^2 + x + 1} }$

Tips and Tricks

• Make sure to write the dividend and divisor in the correct format.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the divisor by the quotient obtained in the previous step.
• Subtract the product from the dividend.
• Bring down the next term of the dividend.
• Repeat the process until the remainder is zero.

Common Mistakes

• Not writing the dividend and divisor in the correct format.
• Not dividing the leading term of the dividend by the leading term of the divisor.
• Not multiplying the divisor by the quotient obtained in the previous step.
• Not subtracting the product from the dividend.
• Not bringing down the next term of the dividend.

Conclusion

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Introduction

In our previous article, we learned how to divide a polynomial using long division. Long division is a method of dividing a polynomial by another polynomial of lower degree. It is a powerful tool in algebra that helps us simplify complex expressions and solve equations. In this article, we will answer some frequently asked questions about long division.

Q: What is long division in algebra?

A: Long division in algebra is a method of dividing a polynomial by another polynomial of lower degree. It is a powerful tool that helps us simplify complex expressions and solve equations.

Q: How do I start long division?

A: To start long division, you need to write the dividend and divisor in the correct format. The dividend is the polynomial that you want to divide, and the divisor is the polynomial by which you are dividing.

Q: What is the first step in long division?

A: The first step in long division is to divide the leading term of the dividend by the leading term of the divisor.

Q: How do I multiply the divisor by the quotient?

A: To multiply the divisor by the quotient, you need to multiply each term of the divisor by the quotient.

Q: How do I subtract the product from the dividend?

A: To subtract the product from the dividend, you need to subtract each term of the product from the corresponding term of the dividend.

Q: What is the remainder in long division?

A: The remainder in long division is the amount left over after you have divided the dividend by the divisor.

Q: How do I write the quotient and remainder?

A: To write the quotient and remainder, you need to write the quotient followed by the remainder divided by the divisor.

Q: What are some common mistakes to avoid in long division?

A: Some common mistakes to avoid in long division include not writing the dividend and divisor in the correct format, not dividing the leading term of the dividend by the leading term of the divisor, not multiplying the divisor by the quotient, and not subtracting the product from the dividend.

Q: How do I practice long division?

A: To practice long division, you can start with simple problems and gradually move on to more complex ones. You can also use online resources and practice problems to help you improve your skills.

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

A: Long division has many real-world applications, including finance, engineering, and science. It is used to solve equations and simplify complex expressions, which is essential in many fields.

Q: Can I use long division to divide fractions?

A: Yes, you can use long division to divide fractions. However, you need to follow the same steps as you would with polynomials.

Q: Can I use long division to divide decimals?

A: Yes, you can use long division to divide decimals. However, you need to follow the same steps as you would with polynomials.

Conclusion

In conclusion, long division is a powerful tool in algebra that helps us simplify complex expressions and solve equations. We answered some frequently asked questions about long division and provided tips and tricks to help you improve your skills. With practice and patience, you can master the art of long division and become proficient in solving complex algebraic expressions.

Example Problems

1. Divide the polynomial $2x^3 + 5x^2 + 3x + 1$ by $x^2 + 2x + 1$.
2. Divide the polynomial $3x^3 - 2x^2 + 4x + 1$ by $x^2 + x + 1$.

Solutions

1. ${ \frac{2x^3 + 5x^2 + 3x + 1}{x^2 + 2x + 1} = 2x + 1 + \frac{3x + 1}{x^2 + 2x + 1} }$
2. ${ \frac{3x^3 - 2x^2 + 4x + 1}{x^2 + x + 1} = 3x - 1 + \frac{5x + 2}{x^2 + x + 1} }$

Tips and Tricks

• Make sure to write the dividend and divisor in the correct format.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the divisor by the quotient obtained in the previous step.
• Subtract the product from the dividend.
• Bring down the next term of the dividend.
• Repeat the process until the remainder is zero.

Common Mistakes

• Not writing the dividend and divisor in the correct format.
• Not dividing the leading term of the dividend by the leading term of the divisor.
• Not multiplying the divisor by the quotient.
• Not subtracting the product from the dividend.
• Not bringing down the next term of the dividend.