Setting the Result Equal to 3

Oct 13, 2025 by miodeya.com 133 views

When a r (x) = -2x^3+kx2 -x+1 is divided by x -2 the rest is 3, then k =?

In algebra, polynomial division is a process of dividing a polynomial by another polynomial. The remainder theorem is a fundamental concept in polynomial division, which states that if a polynomial f(x) is divided by x - a, then the remainder is f(a). In this article, we will use the remainder theorem to find the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3.

The remainder theorem is a powerful tool in algebra that allows us to find the remainder of a polynomial division without actually performing the division. The theorem states that if a polynomial f(x) is divided by x - a, then the remainder is f(a). In other words, we can find the remainder by substituting the value of a into the polynomial f(x).

In this problem, we are given the polynomial r(x) = -2x^3+kx2 -x+1 and we are asked to find the value of k when it is divided by x - 2, given that the remainder is 3. To apply the remainder theorem, we need to substitute x = 2 into the polynomial r(x) and set the result equal to 3.

Substituting x = 2 into the Polynomial

Substituting x = 2 into the polynomial r(x) = -2x^3+kx2 -x+1, we get:

r(2) = -2(2)^3+k(2)2 -2+1 r(2) = -2(8)+4k-2+1 r(2) = -16+4k-1 r(2) = -17+4k

Setting the Result Equal to 3

We are given that the remainder is 3, so we set the result equal to 3:

-17+4k = 3

Solving for k

To solve for k, we need to isolate k on one side of the equation. We can do this by adding 17 to both sides of the equation:

4k = 3+17 4k = 20

Next, we can divide both sides of the equation by 4 to solve for k:

k = 20/4 k = 5

In this article, we used the remainder theorem to find the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3. By substituting x = 2 into the polynomial and setting the result equal to 3, we were able to solve for k and find that k = 5.

The remainder theorem is a powerful tool in algebra that allows us to find the remainder of a polynomial division without actually performing the division.
To apply the remainder theorem, we need to substitute the value of a into the polynomial f(x).
In this problem, we used the remainder theorem to find the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3.
By substituting x = 2 into the polynomial and setting the result equal to 3, we were able to solve for k and find that k = 5.

If you want to learn more about the remainder theorem and polynomial division, I recommend checking out the following resources:

Khan Academy: Polynomial Division
Mathway: Remainder Theorem
Wolfram Alpha: Polynomial Division

I hope this article has been helpful in understanding the remainder theorem and how to apply it to find the value of k in a polynomial division problem.
Q&A: When a r (x) = -2x^3+kx2 -x+1 is divided by x -2 the rest is 3, then k =?

We have received many questions from readers about the problem of finding the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3. Here are some of the most frequently asked questions and their answers:

Q: What is the remainder theorem?

A: The remainder theorem is a powerful tool in algebra that allows us to find the remainder of a polynomial division without actually performing the division. It states that if a polynomial f(x) is divided by x - a, then the remainder is f(a).

Q: How do I apply the remainder theorem to find the value of k?

A: To apply the remainder theorem, you need to substitute the value of a into the polynomial f(x). In this problem, we substituted x = 2 into the polynomial r(x) = -2x^3+kx2 -x+1 and set the result equal to 3.

Q: Why do I need to substitute x = 2 into the polynomial?

A: We need to substitute x = 2 into the polynomial because the problem states that the polynomial is divided by x - 2. This means that we are looking for the remainder when x = 2.

Q: How do I solve for k?

A: To solve for k, we need to isolate k on one side of the equation. We can do this by adding 17 to both sides of the equation and then dividing both sides by 4.

Q: What is the value of k?

A: The value of k is 5.

Q: Can I use the remainder theorem to find the value of k in any polynomial?

A: Yes, you can use the remainder theorem to find the value of k in any polynomial. However, you need to make sure that the polynomial is divided by a linear factor, such as x - a.

Q: What are some common mistakes to avoid when using the remainder theorem?

A: Some common mistakes to avoid when using the remainder theorem include:

Not substituting the correct value of a into the polynomial
Not setting the result equal to the correct remainder
Not isolating k on one side of the equation
Not checking for any errors in the calculation

Q: Where can I learn more about the remainder theorem and polynomial division?

A: You can learn more about the remainder theorem and polynomial division by checking out the following resources:

Khan Academy: Polynomial Division
Mathway: Remainder Theorem
Wolfram Alpha: Polynomial Division

We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about the problem of finding the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3. If you have any further questions, please don't hesitate to ask.

The remainder theorem is a powerful tool in algebra that allows us to find the remainder of a polynomial division without actually performing the division.
To apply the remainder theorem, you need to substitute the value of a into the polynomial f(x).
In this problem, we used the remainder theorem to find the value of k in the polynomial r(x) = -2x^3+kx2 -x+1 when it is divided by x - 2, given that the remainder is 3.
By substituting x = 2 into the polynomial and setting the result equal to 3, we were able to solve for k and find that k = 5.
You can use the remainder theorem to find the value of k in any polynomial, but you need to make sure that the polynomial is divided by a linear factor, such as x - a.