Polynomial Division Finding The Remainder Of (x^3 + 1) Divided By (x^2 - X + 1)
Polynomial division is a fundamental concept in algebra, and understanding how to find remainders is crucial for various mathematical applications. This article will delve into the process of finding the remainder when the polynomial is divided by . We'll explore the underlying principles, the step-by-step method of polynomial long division, and ultimately arrive at the solution. This exploration will not only help you solve this specific problem but also equip you with the knowledge to tackle similar problems in the future. Mastering polynomial division is an essential skill for anyone pursuing further studies in mathematics, engineering, or related fields. It's a building block for more advanced topics such as factoring polynomials, solving equations, and working with rational functions. So, let's embark on this journey of understanding polynomial division and discover the elegant solution to our problem.
The Principles of Polynomial Division
Before we dive into the specifics of our problem, let's lay the foundation by understanding the principles of polynomial division. Polynomial division is analogous to long division with numbers, but instead of dealing with digits, we're working with terms containing variables and exponents. The goal is the same: to divide one polynomial (the dividend) by another (the divisor) and find the quotient and the remainder.
At its core, polynomial division relies on the following equation:
Dividend = (Divisor × Quotient) + Remainder
This equation tells us that if we multiply the divisor by the quotient and add the remainder, we should get back the original dividend. The remainder is the polynomial left over after the division process, and its degree must be less than the degree of the divisor. This is a crucial point to remember. If the degree of the remainder is equal to or greater than the degree of the divisor, we can continue the division process.
In our case, the dividend is and the divisor is . The degree of the dividend is 3, and the degree of the divisor is 2. Therefore, the degree of the remainder must be less than 2, meaning it can be a linear expression (like ) or a constant. Understanding this principle helps us anticipate the form of the remainder and guides us through the division process.
The process of polynomial long division involves systematically dividing the leading term of the dividend by the leading term of the divisor, multiplying the quotient term by the entire divisor, subtracting the result from the dividend, and bringing down the next term. This process is repeated until the degree of the remaining polynomial is less than the degree of the divisor. The remaining polynomial is then the remainder. This step-by-step approach ensures that we accurately account for all terms and arrive at the correct quotient and remainder. By mastering this method, you'll gain a powerful tool for simplifying complex polynomial expressions and solving a wide range of algebraic problems.
Step-by-Step Solution: Dividing (x^3 + 1) by (x^2 - x + 1)
Now, let's apply the principles of polynomial division to our specific problem: dividing by . We'll use the long division method, which provides a clear and organized way to perform the division.
Step 1: Set up the long division.
Write the dividend inside the division symbol and the divisor outside. It's helpful to include placeholder terms with zero coefficients for any missing powers of in the dividend. In this case, we can rewrite as . This ensures proper alignment during the subtraction steps.
_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
Step 2: Divide the leading terms.
Divide the leading term of the dividend () by the leading term of the divisor (). The result is , which becomes the first term of the quotient. Write above the division symbol, aligned with the term in the dividend.
x_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
Step 3: Multiply the quotient term by the divisor.
Multiply the quotient term () by the entire divisor . This gives us . Write this result below the dividend, aligning like terms.
x_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
x^3 - x^2 + x
Step 4: Subtract.
Subtract the expression we just wrote () from the corresponding terms in the dividend . This gives us:
Write the result below the line.
x_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
x^3 - x^2 + x
-----------
x^2 - x + 1
Step 5: Repeat the process.
Now, we treat the result of the subtraction () as the new dividend. Divide the leading term () by the leading term of the divisor (). The result is 1, which becomes the next term in the quotient. Write +1 next to the in the quotient.
x + 1_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
x^3 - x^2 + x
-----------
x^2 - x + 1
Multiply the new quotient term (1) by the divisor . This gives us . Write this below the new dividend.
x + 1_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
x^3 - x^2 + x
-----------
x^2 - x + 1
x^2 - x + 1
Subtract again:
x + 1_________
x^2-x+1 | x^3 + 0x^2 + 0x + 1
x^3 - x^2 + x
-----------
x^2 - x + 1
x^2 - x + 1
-----------
0
Step 6: Identify the remainder.
Since the result of the subtraction is 0, the remainder is 0.
Therefore, when is divided by , the remainder is 0. This means that divides evenly. This result can also be verified by factoring as . This step-by-step breakdown demonstrates the power of polynomial long division in simplifying expressions and revealing important relationships between polynomials. By carefully following each step, you can confidently tackle similar division problems and gain a deeper understanding of algebraic manipulations.
Factoring as an Alternative Method
While polynomial long division is a reliable method for finding remainders, there's often a more elegant and efficient approach: factoring. In this particular case, factoring the dividend provides a direct path to the solution. Recognizing common factoring patterns can significantly simplify polynomial problems and save you valuable time.
The expression is a classic example of the sum of cubes. The sum of cubes factorization formula is:
Applying this formula to our dividend, where and , we get:
Now, we can clearly see that can be factored into and . This factorization immediately reveals that is a factor of . Therefore, when is divided by , the quotient is and the remainder is 0.
This approach highlights the importance of recognizing and applying factoring patterns. Factoring not only simplifies the division process but also provides insights into the relationships between polynomials. In many cases, factoring can be a quicker and more intuitive method than long division. By developing your factoring skills, you'll gain a powerful tool for solving a wide range of algebraic problems and deepening your understanding of polynomial behavior. The ability to recognize patterns and apply appropriate techniques is a hallmark of mathematical proficiency, and factoring is a prime example of this skill.
Conclusion: The Remainder is 0
In conclusion, we've explored two methods for finding the remainder when is divided by : polynomial long division and factoring. Both methods lead us to the same answer: the remainder is 0.
Polynomial long division provided a systematic, step-by-step approach to the problem, reinforcing the fundamental principles of polynomial division. We carefully divided the leading terms, multiplied the quotient term by the divisor, subtracted, and repeated the process until we reached a remainder of 0. This method is valuable for tackling more complex polynomial division problems where factoring may not be readily apparent.
Alternatively, recognizing the sum of cubes pattern allowed us to factor as . This factorization immediately revealed that is a factor of , making the remainder 0. This approach highlights the importance of mastering factoring techniques for simplifying polynomial expressions and solving problems efficiently.
The fact that the remainder is 0 indicates that divides evenly. This means that is a factor of , a relationship that is readily apparent from the factorization. Understanding these relationships between polynomials is crucial for solving equations, simplifying expressions, and tackling more advanced mathematical concepts.
Therefore, the correct answer is D. 0. This exercise demonstrates the power of both polynomial long division and factoring in solving algebraic problems. By mastering these techniques, you'll be well-equipped to tackle a wide range of polynomial-related challenges and deepen your understanding of algebraic principles. The ability to choose the most appropriate method for a given problem is a key skill in mathematics, and this example illustrates the value of having multiple tools in your problem-solving arsenal.