The Result Of The Division (3x² + 4x - 4) Is Divided By (2x + 4)
Introduction
In the world of mathematics, particularly in algebra, division of polynomials is a crucial operation that helps in solving equations and finding the roots of polynomials. In this article, we will discuss the division of the polynomial (3x² + 4x - 4) by (2x + 4) using the long division method. This operation is essential in various fields, including physics, engineering, and computer science.
Understanding the Polynomials
Before we proceed with the division, let's understand the two polynomials involved. The dividend is (3x² + 4x - 4), and the divisor is (2x + 4). The dividend is a quadratic polynomial, while the divisor is a linear polynomial.
Dividend: 3x² + 4x - 4
The dividend is a quadratic polynomial of the form ax² + bx + c, where a = 3, b = 4, and c = -4. This polynomial can be factored as (3x - 1)(x + 4).
Divisor: 2x + 4
The divisor is a linear polynomial of the form mx + n, where m = 2 and n = 4.
Long Division Method
To divide the dividend by the divisor, we will use the long division method. This method involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.
Step 1: Divide the Leading Term
The leading term of the dividend is 3x², and the leading term of the divisor is 2x. To divide 3x² by 2x, we get (3/2)x.
Step 2: Multiply the Divisor
We multiply the entire divisor (2x + 4) by (3/2)x, which gives us (3x + 6).
Step 3: Subtract
We subtract (3x + 6) from the dividend (3x² + 4x - 4), which gives us (3x² - 3x - 10).
Step 4: Bring Down the Next Term
We bring down the next term, which is -10.
Step 5: Repeat the Process
We repeat the process by dividing the leading term of the new dividend (-3x) by the leading term of the divisor (2x), which gives us (-3/2).
Step 6: Multiply the Divisor
We multiply the entire divisor (2x + 4) by (-3/2), which gives us (-3x - 6).
Step 7: Subtract
We subtract (-3x - 6) from the new dividend (-3x - 10), which gives us 4.
Step 8: Write the Result
The result of the division is (3/2)x - 3/2 + 4/(2x + 4).
Simplifying the Result
We can simplify the result by combining the terms. The result can be written as (3/2)x - 3/2 + 4/(2x + 4).
Combining the Terms
We can combine the terms by finding a common denominator. The common denominator is 2x + 4.
Simplifying the Terms
We can simplify the terms by multiplying the numerator and denominator of each term by the necessary factors.
Final Result
The final result of the division is (3/2)x - 3/2 + 2.
Conclusion
In this article, we discussed the division of the polynomial (3x² + 4x - 4) by (2x + 4) using the long division method. We broke down the process into steps and simplified the result. This operation is essential in various fields, including physics, engineering, and computer science.
Introduction
In our previous article, we discussed the division of the polynomial (3x² + 4x - 4) by (2x + 4) using the long division method. In this article, we will answer some frequently asked questions related to this topic.
Q&A
Q: What is the result of the division (3x² + 4x - 4) by (2x + 4)?
A: The result of the division is (3/2)x - 3/2 + 2.
Q: How do I simplify the result of the division?
A: You can simplify the result by combining the terms. The common denominator is 2x + 4. You can simplify the terms by multiplying the numerator and denominator of each term by the necessary factors.
Q: What is the difference between the dividend and the divisor?
A: The dividend is the polynomial being divided, which is (3x² + 4x - 4). The divisor is the polynomial by which we are dividing, which is (2x + 4).
Q: What is the purpose of the long division method?
A: The long division method is used to divide polynomials. It involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.
Q: Can I use the long division method to divide any polynomial?
A: Yes, you can use the long division method to divide any polynomial. However, the divisor must be a linear polynomial.
Q: What is the significance of the result of the division?
A: The result of the division is used to solve equations and find the roots of polynomials. It is also used in various fields, including physics, engineering, and computer science.
Q: How do I check my work when using the long division method?
A: You can check your work by multiplying the entire divisor by the result and subtracting it from the dividend. If the result is zero, then your work is correct.
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 multiplying the entire divisor by the result
- Not subtracting the result from the dividend
- Not checking your work
- Not simplifying the result
Conclusion
In this article, we answered some frequently asked questions related to the division of the polynomial (3x² + 4x - 4) by (2x + 4) using the long division method. We hope that this article has been helpful in clarifying any doubts you may have had.
Additional Resources
If you are looking for additional resources to help you understand the division of polynomials, we recommend the following:
- Khan Academy: Polynomial Division
- Mathway: Polynomial Division
- Wolfram Alpha: Polynomial Division
We hope that these resources will be helpful in your studies.
Final Thoughts
The division of polynomials is an essential operation in mathematics, particularly in algebra. It is used to solve equations and find the roots of polynomials. In this article, we discussed the division of the polynomial (3x² + 4x - 4) by (2x + 4) using the long division method. We hope that this article has been helpful in clarifying any doubts you may have had.