Multiply the divisor by the result

Oct 13, 2025 by windailys.com 66 views

Introduction

Long division is a mathematical technique used to divide polynomials, which are expressions consisting of variables and coefficients. In this article, we will explore how to use long division to find the quotient of the polynomial (3n to the fourth + 6n to the third - 15n to the second + 32n - 25) divided by (n + 4). This process 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.

Understanding the Problem

To begin, we need to understand the concept of long division in algebra. When dividing polynomials, we are essentially finding the quotient and remainder of the division. The quotient is the result of the division, while the remainder is the amount left over after the division is complete. In this case, we are dividing the polynomial (3n to the fourth + 6n to the third - 15n to the second + 32n - 25) by (n + 4).

Step 1: Divide the Highest Degree Term

The first step in long division is to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is 3n to the fourth, and the highest degree term of the divisor is n. To divide 3n to the fourth by n, we can use the rule that n to the power of a divided by n to the power of b is equal to n to the power of a-b. Therefore, 3n to the fourth divided by n is equal to 3n to the third.

Step 2: Multiply the Divisor by the Result

Next, we multiply the entire divisor by the result from step 1. In this case, we multiply (n + 4) by 3n to the third. This gives us 3n to the fourth + 12n to the third.

Step 3: Subtract the Result from the Dividend

Now, we subtract the result from step 2 from the dividend. In this case, we subtract (3n to the fourth + 12n to the third) from (3n to the fourth + 6n to the third - 15n to the second + 32n - 25). This gives us -6n to the third - 15n to the second + 32n - 25.

Step 4: Repeat the Process

We repeat the process of dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we divide -6n to the third by n, which gives us -6n to the second. We then multiply (n + 4) by -6n to the second, which gives us -6n to the third - 24n to the second. We subtract this result from the dividend, which gives us 7n to the second + 32n - 25.

Step 5: Repeat the Process Again

We repeat the process again, dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we divide 7n to the second by n, which gives us 7n. We then multiply (n + 4) by 7n, which gives us 7n to the second + 28n. We subtract this result from the dividend, which gives us 4n - 25.

Step 6: Repeat the Process Again

We repeat the process again, dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we divide 4n by n, which gives us 4. We then multiply (n + 4) by 4, which gives us 4n + 16. We subtract this result from the dividend, which gives us -41.

Conclusion

In conclusion, the quotient of the polynomial (3n to the fourth + 6n to the third - 15n to the second + 32n - 25) divided by (n + 4) is 3n to the third - 6n to the second + 7n - 4. The remainder is -41.

Example Use Case

This technique of long division in algebra can be used in a variety of real-world applications, such as:

Engineering: When designing electrical circuits, engineers may need to divide polynomials to find the transfer function of a circuit.
Computer Science: In computer graphics, programmers may need to divide polynomials to find the equation of a curve.
Physics: In physics, researchers may need to divide polynomials to find the equation of motion of an object.

Tips and Tricks

Here are some tips and tricks to help you master long division in algebra:

Use the rule of exponents: When dividing polynomials, use the rule of exponents to simplify the expression.
Multiply the divisor by the result: When multiplying the divisor by the result, make sure to multiply every term in the divisor by the result.
Subtract the result from the dividend: When subtracting the result from the dividend, make sure to subtract every term in the result from the dividend.

Conclusion

In conclusion, long division in algebra is a powerful technique used to divide polynomials. By following the steps outlined in this article, you can master the technique of long division in algebra and apply it to a variety of real-world applications. Remember to use the rule of exponents, multiply the divisor by the result, and subtract the result from the dividend. With practice and patience, you can become proficient in long division in algebra.

Q: What is long division in algebra?

A: Long division in algebra is a mathematical technique used to divide polynomials, which are expressions consisting of variables and coefficients. 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: Why is long division in algebra important?

A: Long division in algebra is important because it allows us to divide polynomials, which is a fundamental operation in algebra. It has many real-world applications, such as in engineering, computer science, and physics.

Q: How do I start long division in algebra?

A: To start long division in algebra, you need to divide the highest degree term of the dividend by the highest degree term of the divisor. This will give you the first term of the quotient.

Q: What is the remainder in long division in algebra?

A: The remainder in long division in algebra is the amount left over after the division is complete. It is the difference between the dividend and the product of the divisor and the quotient.

Q: How do I find the remainder in long division in algebra?

A: To find the remainder in long division in algebra, you need to subtract the product of the divisor and the quotient from the dividend.

Q: What is the quotient in long division in algebra?

A: The quotient in long division in algebra is the result of the division. It is the product of the divisor and the quotient.

Q: How do I find the quotient in long division in algebra?

A: To find the quotient in long division in algebra, you need to divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result and subtract it from the dividend.

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

A: Some common mistakes to avoid in long division in algebra include:

Not using the rule of exponents: When dividing polynomials, use the rule of exponents to simplify the expression.
Not multiplying the divisor by the result: When multiplying the divisor by the result, make sure to multiply every term in the divisor by the result.
Not subtracting the result from the dividend: When subtracting the result from the dividend, make sure to subtract every term in the result from the dividend.

Q: How can I practice long division in algebra?

A: You can practice long division in algebra by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

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

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

Engineering: When designing electrical circuits, engineers may need to divide polynomials to find the transfer function of a circuit.
Computer Science: In computer graphics, programmers may need to divide polynomials to find the equation of a curve.
Physics: In physics, researchers may need to divide polynomials to find the equation of motion of an object.

Q: How can I use long division in algebra in my career?

A: You can use long division in algebra in your career by applying it to real-world problems in engineering, computer science, and physics. You can also use it to solve problems in other fields, such as economics and finance.