Find The Remainder When $x^3-2$ Is Divided By $x-1$. What Is The Remainder?A. $-x^2-2$ B. $x^2-2$ C. $-2$ D. $-1$

Feb 26, 2025 by ADMIN 117 views

**Finding the Remainder of a Polynomial Division**

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Introduction

When a polynomial is divided by another polynomial, the remainder is a polynomial of degree less than the divisor. In this problem, we are given a polynomial $x^3-2$ and asked to find the remainder when it is divided by $x-1$ . This is a classic problem in algebra, and we will use the Remainder Theorem to solve it.

The Remainder Theorem

The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x-a$ , then the remainder is equal to $f(a)$ . In other words, if we want to find the remainder of a polynomial when it is divided by a linear factor, we can simply evaluate the polynomial at the value of the factor.

Applying the Remainder Theorem

In this problem, we want to find the remainder of the polynomial $x^3-2$ when it is divided by $x-1$ . Using the Remainder Theorem, we can evaluate the polynomial at $x=1$ to find the remainder.

Evaluating the Polynomial

To evaluate the polynomial $x^3-2$ at $x=1$ , we simply substitute $x=1$ into the polynomial:

f(1) = (1)^3 - 2 = 1 - 2 = -1

Conclusion

Therefore, the remainder of the polynomial $x^3-2$ when it is divided by $x-1$ is $-1$ . This is the correct answer, and we can see that it is option D.

Why the Other Options are Incorrect

Let's take a look at the other options to see why they are incorrect.

Option A: $-x^2-2$ is not the correct remainder because it is not equal to $f(1)$ .
Option B: $x^2-2$ is not the correct remainder because it is not equal to $f(1)$ .
Option C: $-2$ is not the correct remainder because it is not equal to $f(1)$ .

Example Use Case

The Remainder Theorem is a powerful tool that can be used to solve a wide range of problems in algebra. For example, suppose we want to find the remainder of the polynomial $x^2+3x-4$ when it is divided by $x+2$ . We can use the Remainder Theorem to evaluate the polynomial at $x=-2$ :

f(-2) = (-2)^2 + 3(-2) - 4 = 4 - 6 - 4 = -6

Therefore, the remainder of the polynomial $x^2+3x-4$ when it is divided by $x+2$ is $-6$ .

Conclusion

In conclusion, the Remainder Theorem is a powerful tool that can be used to solve a wide range of problems in algebra. By evaluating a polynomial at a specific value, we can find the remainder of the polynomial when it is divided by a linear factor. In this problem, we used the Remainder Theorem to find the remainder of the polynomial $x^3-2$ when it is divided by $x-1$ , and we found that the remainder is $-1$ .

Final Answer

The final answer is $\boxed{-1}$ .

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Q: What is the Remainder Theorem?

A: The Remainder Theorem is a fundamental concept in algebra that states that if a polynomial $f(x)$ is divided by $x-a$ , then the remainder is equal to $f(a)$ . This means that we can find the remainder of a polynomial by evaluating the polynomial at the value of the divisor.

Q: How do I apply the Remainder Theorem?

A: To apply the Remainder Theorem, you need to follow these steps:

Identify the polynomial and the divisor.
Evaluate the polynomial at the value of the divisor.
The result is the remainder.

Q: What are some common mistakes to avoid when using the Remainder Theorem?

A: Here are some common mistakes to avoid when using the Remainder Theorem:

Not evaluating the polynomial at the correct value.
Not simplifying the expression before evaluating it.
Not checking if the divisor is a factor of the polynomial.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial when it is divided by a quadratic factor?

A: No, the Remainder Theorem only works for linear factors. If you want to find the remainder of a polynomial when it is divided by a quadratic factor, you need to use a different method, such as polynomial long division.

Q: How do I use the Remainder Theorem to solve a problem?

A: Here's an example of how to use the Remainder Theorem to solve a problem:

Suppose we want to find the remainder of the polynomial $x^2+3x-4$ when it is divided by $x+2$ . We can use the Remainder Theorem to evaluate the polynomial at $x=-2$ :

f(-2) = (-2)^2 + 3(-2) - 4 = 4 - 6 - 4 = -6

Therefore, the remainder of the polynomial $x^2+3x-4$ when it is divided by $x+2$ is $-6$ .

Q: What are some real-world applications of the Remainder Theorem?

A: The Remainder Theorem has many real-world applications, including:

Computer science: The Remainder Theorem is used in computer science to find the remainder of a polynomial when it is divided by a linear factor.
Engineering: The Remainder Theorem is used in engineering to find the remainder of a polynomial when it is divided by a linear factor.
Economics: The Remainder Theorem is used in economics to find the remainder of a polynomial when it is divided by a linear factor.

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

A: No, the Remainder Theorem only works for linear factors. If you want to find the remainder of a polynomial when it is divided by a polynomial of degree greater than 1, you need to use a different method, such as polynomial long division.

Q: How do I know if the Remainder Theorem is applicable to a problem?

A: The Remainder Theorem is applicable to a problem if the divisor is a linear factor. If the divisor is a polynomial of degree greater than 1, you need to use a different method, such as polynomial long division.

Conclusion

In conclusion, the Remainder Theorem is a powerful tool that can be used to find the remainder of a polynomial when it is divided by a linear factor. By following the steps outlined in this article, you can use the Remainder Theorem to solve a wide range of problems in algebra.

Final Answer

The final answer is $\boxed{0}$ .