What Is The Remainder When $x^3 + 1$ Is Divided By $x^2 - X + 1$?A. \$x + 1$[/tex\] B. $x$ C. 2 D. 0

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What is the Remainder When $x^3 + 1$ is Divided by $x^2 - x + 1$?

In algebra, the remainder theorem is a fundamental concept used to find the remainder of a polynomial when divided by another polynomial. Given two polynomials, $f(x)$ and $g(x)$, the remainder theorem states that if $f(x)$ is divided by $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$. In this article, we will explore the remainder when $x^3 + 1$ is divided by $x^2 - x + 1$.

The remainder theorem can be used to find the remainder of a polynomial when divided by another polynomial. To do this, we need to perform polynomial long division. However, in this case, we can use the fact that the remainder is a polynomial of degree less than the degree of the divisor.

To find the remainder when $x^3 + 1$ is divided by $x^2 - x + 1$, we can use polynomial long division. We start by dividing the highest degree term of the dividend by the highest degree term of the divisor.

x3x2=x\frac{x^3}{x^2} = x

We multiply the divisor by $x$ and subtract it from the dividend.

x3+1−(x3−x2+x)=x2−x+1x^3 + 1 - (x^3 - x^2 + x) = x^2 - x + 1

We repeat the process by dividing the highest degree term of the result by the highest degree term of the divisor.

x2x2=1\frac{x^2}{x^2} = 1

We multiply the divisor by $1$ and subtract it from the result.

x2−x+1−(x2−x+1)=0x^2 - x + 1 - (x^2 - x + 1) = 0

Since the result is $0$, we have found the remainder.

The remainder when $x^3 + 1$ is divided by $x^2 - x + 1$ is $0$. This means that $x^3 + 1$ is exactly divisible by $x^2 - x + 1$.

In conclusion, the remainder theorem can be used to find the remainder of a polynomial when divided by another polynomial. In this article, we used polynomial long division to find the remainder when $x^3 + 1$ is divided by $x^2 - x + 1$. We found that the remainder is $0$, which means that $x^3 + 1$ is exactly divisible by $x^2 - x + 1$.

The final answer is D. 0.

  • The remainder theorem can be used to find the remainder of a polynomial when divided by another polynomial.
  • Polynomial long division can be used to find the remainder of a polynomial when divided by another polynomial.
  • The remainder is a polynomial of degree less than the degree of the divisor.
  • [1] "The Remainder Theorem" by Math Open Reference
  • [2] "Polynomial Long Division" by Math Is Fun
  • The Remainder Theorem
  • Polynomial Long Division
  • Algebra
  • Q: What is the remainder when $x^3 + 1$ is divided by $x^2 - x + 1$? A: The remainder is $0$.
  • Q: How do I find the remainder of a polynomial when divided by another polynomial? A: You can use the remainder theorem or polynomial long division.
  • Q: What is the remainder theorem? A: The remainder theorem is a fundamental concept used to find the remainder of a polynomial when divided by another polynomial.
    Frequently Asked Questions: The Remainder Theorem and Polynomial Long Division ====================================================================================

In our previous article, we explored the remainder when $x^3 + 1$ is divided by $x^2 - x + 1$. We used the remainder theorem and polynomial long division to find the remainder. In this article, we will answer some frequently asked questions related to the remainder theorem and polynomial long division.

Q: What is the remainder theorem?

A: The remainder theorem is a fundamental concept used to find the remainder of a polynomial when divided by another polynomial. It states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$.

Q: How do I find the remainder of a polynomial when divided by another polynomial?

A: You can use the remainder theorem or polynomial long division. The remainder theorem is a more general method that can be used to find the remainder of a polynomial when divided by another polynomial, while polynomial long division is a more specific method that can be used to find the remainder of a polynomial when divided by a linear polynomial.

Q: What is polynomial long division?

A: Polynomial long division is a method used to divide a polynomial by another polynomial. It is similar to long division, but it is used for polynomials instead of integers. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the divisor by the result and subtracting it from the dividend.

Q: How do I perform polynomial long division?

A: To perform polynomial long division, you need to follow these steps:

  1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
  2. Multiply the divisor by the result and subtract it from the dividend.
  3. Repeat the process until the degree of the dividend is less than the degree of the divisor.

Q: What is the remainder of a polynomial when divided by a linear polynomial?

A: The remainder of a polynomial when divided by a linear polynomial is a constant. This is because a linear polynomial has only one term, and when you divide a polynomial by a linear polynomial, the remainder is the constant term of the polynomial.

Q: Can I use the remainder theorem to find the remainder of a polynomial when divided by a linear polynomial?

A: Yes, you can use the remainder theorem to find the remainder of a polynomial when divided by a linear polynomial. The remainder theorem states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$. Since a linear polynomial has degree 1, the remainder will be a constant.

Q: What is the difference between the remainder theorem and polynomial long division?

A: The remainder theorem is a more general method that can be used to find the remainder of a polynomial when divided by another polynomial, while polynomial long division is a more specific method that can be used to find the remainder of a polynomial when divided by a linear polynomial. The remainder theorem can be used to find the remainder of a polynomial when divided by a polynomial of any degree, while polynomial long division is only used for polynomials of degree 1.

Q: Can I use polynomial long division to find the remainder of a polynomial when divided by a polynomial of degree greater than 1?

A: No, you cannot use polynomial long division to find the remainder of a polynomial when divided by a polynomial of degree greater than 1. Polynomial long division is only used for polynomials of degree 1, while polynomial long division can be used to find the remainder of a polynomial when divided by a polynomial of degree greater than 1.

Q: What is the remainder of a polynomial when divided by a polynomial of degree greater than 1?

A: The remainder of a polynomial when divided by a polynomial of degree greater than 1 is a polynomial of degree less than the degree of the divisor. This is because the remainder theorem states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$.

Q: Can I use the remainder theorem to find the remainder of a polynomial when divided by a polynomial of degree greater than 1?

A: Yes, you can use the remainder theorem to find the remainder of a polynomial when divided by a polynomial of degree greater than 1. The remainder theorem states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$. Since the degree of the divisor is greater than 1, the remainder will be a polynomial of degree less than the degree of the divisor.

Q: What is the difference between the remainder theorem and synthetic division?

A: The remainder theorem is a more general method that can be used to find the remainder of a polynomial when divided by another polynomial, while synthetic division is a specific method that can be used to find the remainder of a polynomial when divided by a linear polynomial. The remainder theorem can be used to find the remainder of a polynomial when divided by a polynomial of any degree, while synthetic division is only used for polynomials of degree 1.

Q: Can I use synthetic division to find the remainder of a polynomial when divided by a polynomial of degree greater than 1?

A: No, you cannot use synthetic division to find the remainder of a polynomial when divided by a polynomial of degree greater than 1. Synthetic division is only used for polynomials of degree 1, while synthetic division can be used to find the remainder of a polynomial when divided by a polynomial of degree greater than 1.

Q: What is the remainder of a polynomial when divided by a polynomial of degree greater than 1 using synthetic division?

A: The remainder of a polynomial when divided by a polynomial of degree greater than 1 using synthetic division is a polynomial of degree less than the degree of the divisor. This is because the remainder theorem states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$.

Q: Can I use the remainder theorem to find the remainder of a polynomial when divided by a polynomial of degree greater than 1 using synthetic division?

A: Yes, you can use the remainder theorem to find the remainder of a polynomial when divided by a polynomial of degree greater than 1 using synthetic division. The remainder theorem states that if a polynomial $f(x)$ is divided by another polynomial $g(x)$, then the remainder is a polynomial of degree less than the degree of $g(x)$. Since the degree of the divisor is greater than 1, the remainder will be a polynomial of degree less than the degree of the divisor.

In conclusion, the remainder theorem and polynomial long division are two important concepts in algebra that can be used to find the remainder of a polynomial when divided by another polynomial. The remainder theorem is a more general method that can be used to find the remainder of a polynomial when divided by a polynomial of any degree, while polynomial long division is a more specific method that can be used to find the remainder of a polynomial when divided by a linear polynomial.