Divide Using Long Division. State The Quotient, $q(x)$, And The Remainder, $r(x)$.$\left(8x^3 - 6x^2 - 21x - 35\right) \div (2x - 5)$$\left(8x^3 - 6x^2 - 21x - 35\right) \div (2x - 5) = \square + \frac{\square}{2x

Introduction

Long division is a mathematical technique used to divide a polynomial by another polynomial. It is an essential tool in algebra and is used to simplify complex expressions. In this article, we will learn how to divide a polynomial using long division and state the quotient, $q(x)$, and the remainder, $r(x)$.

The Division Problem

The division problem we will be using is:

$$\left(8x^3 - 6x^2 - 21x - 35\right) \div (2x - 5)$$

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 being divided, and the divisor is the polynomial by which we are dividing.

	8x^3	-6x^2	-21x	-35
2x - 5

Step 2: Divide the Leading Term

The leading term of the dividend is 8x^3, and the leading term of the divisor is 2x. To divide the leading term of the dividend by the leading term of the divisor, we need to divide 8x^3 by 2x.

$$\frac{8x^3}{2x} = 4x^2$$

Step 3: Multiply the Divisor by the Result

Now that we have the result of the division, we need to multiply the divisor by this result.

$$(2x - 5) \times 4x^2 = 8x^3 - 20x^2$$

Step 4: Subtract the Result from the Dividend

We now subtract the result from the dividend.

$$\left(8x^3 - 6x^2 - 21x - 35\right) - \left(8x^3 - 20x^2\right) = 14x^2 - 21x - 35$$

Step 5: Bring Down the Next Term

We now bring down the next term of the dividend.

	14x^2	-21x	-35
2x - 5

Step 6: Repeat the Process

We now repeat the process of dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend.

$$\frac{14x^2}{2x} = 7x$$

$$(2x - 5) \times 7x = 14x^2 - 35x$$

$$\left(14x^2 - 21x - 35\right) - \left(14x^2 - 35x\right) = 14x - 35$$

Step 7: Bring Down the Next Term

We now bring down the next term of the dividend.

	14x	-35
2x - 5

Step 8: Repeat the Process

We now repeat the process of dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend.

$$\frac{14x}{2x} = 7$$

$$(2x - 5) \times 7 = 14x - 35$$

$$\left(14x - 35\right) - \left(14x - 35\right) = 0$$

The Quotient and Remainder

The quotient of the division is:

$$q(x) = 4x^2 + 7x + 7$$

The remainder of the division is:

$$r(x) = 0$$

Conclusion

In this article, we learned how to divide a polynomial using long division and state the quotient, $q(x)$, and the remainder, $r(x)$. We used the division problem:

$$\left(8x^3 - 6x^2 - 21x - 35\right) \div (2x - 5)$$

to demonstrate the long division process. We divided the leading term of the dividend by the leading term of the divisor, multiplied the divisor by the result, and subtracted the result from the dividend. We repeated this process until we obtained a remainder of 0. The quotient of the division is:

$$q(x) = 4x^2 + 7x + 7$$

and the remainder of the division is:

$$r(x) = 0$$

Example Problems

Here are some example problems to practice long division:

• $$\left(3x^2 + 2x - 1\right) \div (x + 1)$$

• $$\left(2x^3 - 5x^2 + 3x + 1\right) \div (x - 2)$$

• $$\left(x^2 + 4x + 3\right) \div (x + 3)$$

Tips and Tricks

Here are some tips and tricks to help you with long division:

• 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 result.
• Subtract the result from the dividend.
• Repeat the process until you obtain a remainder of 0.

Common Mistakes

Here are some common mistakes to avoid when doing long division:

• 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 result.
• Not subtracting the result from the dividend.
• Not repeating the process until you obtain a remainder of 0.

Conclusion

Q: What is long division?

A: Long division is a mathematical technique used to divide a polynomial by another polynomial. It is an essential tool in algebra and is used to simplify complex expressions.

Q: Why do we need to use long division?

A: We need to use long division to divide polynomials and simplify complex expressions. It is a powerful tool that allows us to break down complex problems into smaller, more manageable parts.

Q: What are the steps involved in long division?

A: The steps involved in long division are:

1. Write the dividend and divisor in the correct format.
2. Divide the leading term of the dividend by the leading term of the divisor.
3. Multiply the divisor by the result.
4. Subtract the result from the dividend.
5. Repeat the process until you obtain a remainder of 0.

Q: What is the quotient and remainder in long division?

A: The quotient is the result of the division, and the remainder is the amount left over after the division is complete.

Q: How do I know when to stop dividing?

A: You know when to stop dividing when you obtain a remainder of 0. This means that the division is complete, and you have found the quotient.

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 result.
• Not subtracting the result from the dividend.
• Not repeating the process until you obtain a remainder of 0.

Q: How can I practice long division?

A: You can practice long division by using online resources, such as math websites and apps, or by working with a tutor or teacher. You can also practice long division by using real-world examples, such as dividing a pizza or a cake.

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

A: Some real-world applications of long division include:

• Cooking and baking: Long division can be used to divide ingredients and measure out the correct amounts.
• Finance: Long division can be used to divide money and calculate interest rates.
• Science: Long division can be used to divide measurements and calculate rates of change.

Q: Can I use long division with fractions?

A: Yes, you can use long division with fractions. However, you will need to use a different method, such as the "invert and multiply" method, to divide fractions.

Q: Can I use long division with decimals?

A: Yes, you can use long division with decimals. However, you will need to use a different method, such as the "rounding" method, to divide decimals.

Q: How can I use long division to solve word problems?

A: You can use long division to solve word problems by breaking down the problem into smaller, more manageable parts. For example, if you are given a word problem that involves dividing a certain number of items into groups, you can use long division to find the number of items in each group.

Q: Can I use long division to solve algebraic equations?

A: Yes, you can use long division to solve algebraic equations. However, you will need to use a different method, such as the "factoring" method, to solve the equation.

Conclusion

Long division is a powerful tool in algebra that allows us to divide polynomials and simplify complex expressions. By following the steps outlined in this article, you can master the art of long division and become proficient in dividing polynomials. Remember 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 result, subtract the result from the dividend, and repeat the process until you obtain a remainder of 0. With practice and patience, you will become a pro at long division and be able to tackle even the most complex division problems.