Remainder Of Polynomial Division X³ + 1 By X² - X + 1

Jul 16, 2025 by ADMIN 54 views

Understanding Polynomial Division and Remainders

Polynomial division is a fundamental concept in algebra, allowing us to divide one polynomial by another. This process is similar to long division with numbers, and it helps us understand the relationship between polynomials, their factors, and remainders. In this comprehensive article, we will dive into the intricacies of polynomial division, specifically focusing on finding the remainder when the polynomial (x³ + 1) is divided by (x² - x + 1). This is a classic problem that highlights the importance of understanding polynomial factorization, the division algorithm, and how to apply these concepts to find the correct remainder. We will explore the steps involved in performing polynomial long division, discuss alternative methods for solving the problem, and delve into the underlying mathematical principles that make this process work. By the end of this article, you will have a solid understanding of how to tackle similar problems and a deeper appreciation for the elegance of polynomial algebra.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing one polynomial by another polynomial of a lower or equal degree. It's a crucial technique in algebra and calculus, enabling us to simplify complex expressions, factor polynomials, and solve equations. Before we tackle the specific problem of dividing (x³ + 1) by (x² - x + 1), let's first review the general process of polynomial long division.

The process mirrors long division with numbers. The dividend is the polynomial being divided (x³ + 1 in our case), and the divisor is the polynomial we're dividing by (x² - x + 1). The goal is to find the quotient and the remainder.

Set up the division: Write the dividend inside the long division symbol and the divisor outside. Make sure to include placeholders (terms with a coefficient of 0) for any missing powers of x in the dividend. For x³ + 1, we can rewrite it as x³ + 0x² + 0x + 1 to make the division process clearer.
Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient. In our case, we divide x³ by x², which gives us x. Write this x above the division symbol.
Multiply: Multiply the entire divisor by the term you just wrote in the quotient (x). This gives you x(x² - x + 1) = x³ - x² + x. Write this result below the dividend, aligning like terms.
Subtract: Subtract the result from the dividend. This step is crucial for finding the next term to bring down. Subtracting (x³ - x² + x) from (x³ + 0x² + 0x + 1) gives us (x² - x + 1).
Bring down: Bring down the next term from the dividend (if there is one) and write it next to the result of the subtraction. In this case, we don't need to bring down any more terms since we already have a polynomial (x² - x + 1).
Repeat: Repeat steps 2-5 with the new polynomial obtained after the subtraction. Divide the leading term of the new polynomial (x²) by the leading term of the divisor (x²). This gives us 1, which is the next term of the quotient. Write +1 next to the x in the quotient.
Multiply and Subtract Again: Multiply the divisor (x² - x + 1) by the new term in the quotient (1). This gives us 1(x² - x + 1) = x² - x + 1. Subtract this from the current polynomial (x² - x + 1). The result is 0. This indicates that the remainder is 0.
Remainder: If the degree of the resulting polynomial is less than the degree of the divisor, then that polynomial is the remainder. If the result is 0, as in this case, the remainder is 0.

Polynomial long division can seem intricate at first, but with practice, it becomes a straightforward process. By understanding each step and applying it methodically, you can confidently divide polynomials and determine both the quotient and the remainder.

Step-by-Step Solution: Dividing (x³ + 1) by (x² - x + 1)

Now that we've reviewed the general process of polynomial long division, let's apply it to the specific problem of dividing (x³ + 1) by (x² - x + 1). This step-by-step solution will demonstrate how to perform the division and identify the remainder.

Set up the division: As mentioned earlier, we rewrite the dividend (x³ + 1) as (x³ + 0x² + 0x + 1) to include placeholders for the missing terms. This helps keep the terms aligned during the division process. Set up the long division as follows:
```
             ________
x² - x + 1 | x³ + 0x² + 0x + 1
```
Divide the leading terms: Divide the leading term of the dividend (x³) by the leading term of the divisor (x²). This gives us x, which is the first term of the quotient. Write x above the division symbol:
```
             x_______
x² - x + 1 | x³ + 0x² + 0x + 1
```
Multiply: Multiply the divisor (x² - x + 1) by the first term of the quotient (x). This gives us x(x² - x + 1) = x³ - x² + x. Write this result below the dividend, aligning like terms:
```
             x_______
x² - x + 1 | x³ + 0x² + 0x + 1
            x³ -  x² +  x
```

Subtract: Subtract the result from the dividend. Subtracting (x³ - x² + x) from (x³ + 0x² + 0x + 1) gives us:

             x_______
x² - x + 1 | x³ + 0x² + 0x + 1
            -(x³ -  x² +  x)
            ------------------
                 x² -  x + 1

Repeat: Now, divide the leading term of the new polynomial (x²) by the leading term of the divisor (x²). This gives us 1, which is the next term of the quotient. Write +1 next to the x in the quotient:
```
             x + 1___
x² - x + 1 | x³ + 0x² + 0x + 1
            -(x³ -  x² +  x)
            ------------------
                 x² -  x + 1
```

Multiply and Subtract Again: Multiply the divisor (x² - x + 1) by the new term in the quotient (1). This gives us 1(x² - x + 1) = x² - x + 1. Subtract this from the current polynomial (x² - x + 1):

             x + 1___
x² - x + 1 | x³ + 0x² + 0x + 1
            -(x³ -  x² +  x)
            ------------------
                 x² -  x + 1
            -(x² -  x + 1)
            ------------------
                     0

Remainder: The result of the subtraction is 0. This means that the remainder is 0. Therefore, when (x³ + 1) is divided by (x² - x + 1), the remainder is 0.

This step-by-step solution clearly illustrates the process of polynomial long division and how it leads to the determination of the remainder. The remainder being 0 indicates that (x² - x + 1) is a factor of (x³ + 1). This will be further explored in the next section.

Alternative Method: Factoring and the Remainder Theorem

While polynomial long division is a reliable method for finding remainders, there are alternative approaches that can sometimes be more efficient, especially when dealing with specific types of polynomials. One such method involves factoring and applying the Remainder Theorem. In this section, we'll explore how these techniques can be used to solve the problem of finding the remainder when (x³ + 1) is divided by (x² - x + 1).

Factoring (x³ + 1)

The first step in this alternative method is to factor the dividend, (x³ + 1). This polynomial is a sum of cubes, which has a well-known factorization formula:

a³ + b³ = (a + b)(a² - ab + b²)

In our case, a = x and b = 1, so we can apply the formula:

x³ + 1 = (x + 1)(x² - x + 1)

This factorization is crucial because it reveals a direct relationship between the dividend (x³ + 1) and the divisor (x² - x + 1). We can see that (x² - x + 1) is one of the factors of (x³ + 1).

Applying the Remainder Theorem

The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is f(c). However, in this case, we are dividing by a quadratic expression, not a linear one. The factored form of (x³ + 1) gives us a more direct approach.

Since x³ + 1 = (x + 1)(x² - x + 1), we can see that (x³ + 1) is perfectly divisible by (x² - x + 1). This means that the division will result in a quotient of (x + 1) and a remainder of 0. This conclusion aligns perfectly with the result we obtained using polynomial long division.

Why This Method Works

The success of this method hinges on the ability to factor the dividend in a way that reveals the divisor as one of its factors. When this occurs, it immediately implies that the remainder will be zero. Factoring simplifies the problem significantly, bypassing the need for the step-by-step process of long division. This approach is particularly useful when the polynomials have readily recognizable factors or when the question is specifically designed to exploit such factorizations.

In summary, by factoring (x³ + 1) and recognizing that (x² - x + 1) is a factor, we can quickly determine that the remainder is 0. This method showcases the power of factorization and its role in simplifying polynomial division problems.

Conclusion: The Remainder is 0

In conclusion, we have explored two methods for determining the remainder when (x³ + 1) is divided by (x² - x + 1): polynomial long division and factoring with the Remainder Theorem. Both approaches have led us to the same answer: the remainder is 0.

Polynomial long division provided a systematic, step-by-step method to perform the division and identify the remainder. This technique is versatile and can be applied to a wide range of polynomial division problems, regardless of the complexity of the polynomials involved. By meticulously following the steps of dividing the leading terms, multiplying, subtracting, and bringing down, we were able to arrive at the remainder of 0.

The alternative method, factoring, offered a more elegant and efficient solution in this particular case. By recognizing that (x³ + 1) can be factored as (x + 1)(x² - x + 1), we immediately saw that (x² - x + 1) is a factor of (x³ + 1). This direct relationship implied that the remainder would be 0, without the need for extensive calculations. This method highlights the importance of recognizing patterns and applying factorization techniques to simplify problems.

Key Takeaways

The remainder when (x³ + 1) is divided by (x² - x + 1) is 0.
Polynomial long division is a reliable method for dividing polynomials and finding remainders.
Factoring can simplify polynomial division problems, especially when the divisor is a factor of the dividend.
Understanding the Remainder Theorem and its implications can lead to efficient problem-solving.

Implications of a Zero Remainder

The fact that the remainder is 0 has significant implications. It means that (x² - x + 1) divides (x³ + 1) evenly, and (x² - x + 1) is a factor of (x³ + 1). This relationship is fundamental in algebra and has applications in various areas, including equation solving, graph sketching, and calculus.

Understanding polynomial division and remainders is not just an exercise in algebraic manipulation; it's a crucial skill for advanced mathematics and problem-solving. The ability to divide polynomials, identify factors, and apply the Remainder Theorem opens doors to more complex mathematical concepts and applications. This article has provided a comprehensive exploration of these concepts, equipping you with the knowledge and skills to tackle similar problems with confidence.