Question

What is the remainder obtained by dividing \(x^{7}+x^{5}+1\) by the generator polynomial \(x^{3}+1 ?\)

Short answer

The remainder is \(-x^2 + x + 1\).

Step-by-step solution

Understanding the Problem

We need to find the remainder when the polynomial \(x^7 + x^5 + 1\) is divided by the polynomial \(x^3 + 1\). This is a polynomial division problem.

Identify the Division

We are dividing a degree 7 polynomial, \(x^7 + x^5 + 1\), by a degree 3 polynomial, \(x^3 + 1\). The remainder will have a degree less than 3 since any polynomial divided by a polynomial of higher degree has a remainder with a lower degree.

Perform Polynomial Long Division

1. Divide the first term: Divide \(x^7\) by \(x^3\) to get \(x^4\).2. Multiply: \(x^4(x^3 + 1) = x^7 + x^4\).3. Subtract from the original: \((x^7 + x^5 + 1) - (x^7 + x^4) = x^5 - x^4 + 1\).4. Repeat step 1: Divide \(x^5\) by \(x^3\) to get \(x^2\).5. Multiply: \(x^2(x^3 + 1) = x^5 + x^2\).6. Subtract: \((x^5 - x^4 + 1) - (x^5 + x^2) = -x^4 - x^2 + 1\).7. Repeat step 1: Divide \(-x^4\) by \(x^3\) to get \(-x\).8. Multiply: \(-x(x^3 + 1) = -x^4 - x\).9. Subtract: \((-x^4 - x^2 + 1) - (-x^4 - x) = -x^2 + x + 1\).10. The degree of the remainder \(-x^2 + x + 1\) is less than 3, so this is the remainder.

Verify the Division and the Remainder

Check that the remainder \(-x^2 + x + 1\) is correct by adding it to the product of the divisor and the quotient and confirming it equals the original polynomial. In this case, \((x^4 + x^2 - x)(x^3 + 1) + (-x^2 + x + 1) = x^7 + x^5 + 1\). This confirms the remainder is correct.

Key concepts

Remainder Theorem

The Remainder Theorem is a crucial concept in polynomial division. It states that when a polynomial, say \(f(x)\), is divided by a linear polynomial of the form \(x-a\), the remainder of the division is \(f(a)\). This concept is particularly useful for quickly finding remainders without having to perform the complete division.

• The theorem applies directly to linear divisors. For non-linear polynomials, traditional division methods like long division are applied.
• For example, when dividing \(f(x) = x^3 + 3x^2 + 3x + 1\) by \(x - 1\), simply substitute \(x = 1\) in \(f(x)\) to find the remainder: \(f(1) = 1^3 + 3 \cdot 1^2 + 3 \cdot 1 + 1 = 8\). Hence, the remainder is 8.

For our exercise, however, the Remainder Theorem is not directly applicable because the divisor \(x^3 + 1\) is not linear. Therefore, we use the long division method to determine the remainder.

Long Division Method

The Long Division Method in polynomials mirrors numeric long division, and helps to break down complex polynomial divisions. Let's dig deeper into this process.

• Identify the terms: Begin by arranging both the dividend and divisor in descending order of degree.
• Divide the leading terms: Start with the highest degree term of the dividend. Divide it by the leading term of the divisor.
• Multiply and subtract: Multiply the entire divisor by the result obtained and subtract from the original polynomial.
• Repeat: Continue this process with the resulting polynomial until you can no longer divide because the degree of the result is less than the divisor.

In our original exercise, we start by dividing \(x^7\) by \(x^3\) to get \(x^4\), and continue this process until the remainder is less than a degree of 3. This method systematically breaks down the division keeping the process organized and efficient.

Polynomial Degree

Understanding the degree of a polynomial is essential when dealing with divisions and remainders. The degree of a polynomial is the highest power of \(x\) with a non-zero coefficient. This degree helps in understanding the complexity and behavior of the polynomial.

• Determining the degree: In a polynomial, terms are typically arranged from the highest degree to the lowest.
• Significance in division: In division, when dividing one polynomial by another, the degree of the quotient and remainder are wholly dependent on the dividend's and divisor's degrees.
• Result importance: After polynomial long division, the degree of the remainder must always be less than that of the divisor, ensuring that the polynomial equation is correctly balanced.

In our example, the divisor \(x^3 + 1\) is degree 3, which means any remainder must have a degree of less than 3. The correct interpretation of polynomial degree is vital to achieve accurate results in polynomial division and verify the correctness of the solution.