Divide the polynomial by the linear factor

Oct 13, 2025 by windailys.com 7 views

**Select the Correct Answer from Each Drop-Down Menu: Using the Remainder Theorem to Verify a Factor**

What is the Remainder Theorem?

The remainder theorem is a fundamental concept in algebra that helps us determine the remainder of a polynomial when it is divided by a linear factor. In this article, we will use the remainder theorem to verify whether a given polynomial is divisible by a specific factor.

What is the Problem Statement?

We are given a polynomial function $f(x) = x^3 + 3x^2 - 25x - 75$ and we need to verify whether $(x+5)$ is a factor of this function using the remainder theorem.

How to Use the Remainder Theorem?

To use the remainder theorem, we need to follow these steps:

Divide the polynomial by the linear factor: In this case, we need to divide $f(x)$ by $(x+5)$ .
Find the remainder: The remainder will be a constant value.
Check if the remainder is zero: If the remainder is zero, then the linear factor is a factor of the polynomial.

Step 1: Divide the Polynomial by the Linear Factor

To divide $f(x)$ by $(x+5)$ , we can use polynomial long division or synthetic division. Let's use polynomial long division.

f(x) = x^3 + 3x^2 - 25x - 75
-------------------------
(x+5) | x^3 + 3x^2 - 25x - 75
      | x^3 + 5x^2
      | ---------
      | -2x^2 - 25x
      | -2x^2 - 10x
      | ---------
      | -15x - 75
      | -15x - 75
      | ---------
      | 0

Step 2: Find the Remainder

From the polynomial long division, we can see that the remainder is zero.

Step 3: Check if the Remainder is Zero

Since the remainder is zero, we can conclude that $(x+5)$ is a factor of the polynomial $f(x)$ .

Q&A

Q: What is the remainder theorem?

A: The remainder theorem is a concept in algebra that helps us determine the remainder of a polynomial when it is divided by a linear factor.

Q: How do I use the remainder theorem?

A: To use the remainder theorem, you need to divide the polynomial by the linear factor, find the remainder, and check if the remainder is zero.

Q: What is the problem statement in this article?

A: The problem statement is to verify whether $(x+5)$ is a factor of the polynomial $f(x) = x^3 + 3x^2 - 25x - 75$ using the remainder theorem.

Q: How do I divide a polynomial by a linear factor?

A: You can use polynomial long division or synthetic division to divide a polynomial by a linear factor.

Q: What is the remainder in this problem?

A: The remainder is zero.

Q: What does it mean if the remainder is zero?

A: If the remainder is zero, then the linear factor is a factor of the polynomial.

Conclusion

In this article, we used the remainder theorem to verify whether $(x+5)$ is a factor of the polynomial $f(x) = x^3 + 3x^2 - 25x - 75$ . We divided the polynomial by the linear factor, found the remainder, and checked if the remainder is zero. Since the remainder is zero, we can conclude that $(x+5)$ is a factor of the polynomial $f(x)$ .

Frequently Asked Questions

Q: What is the remainder theorem used for?

A: The remainder theorem is used to determine the remainder of a polynomial when it is divided by a linear factor.

Q: How do I apply the remainder theorem in real-life situations?

A: You can apply the remainder theorem in real-life situations such as finding the remainder of a polynomial when it is divided by a linear factor, or verifying whether a polynomial is divisible by a specific factor.

Q: What are some common applications of the remainder theorem?

A: Some common applications of the remainder theorem include finding the remainder of a polynomial when it is divided by a linear factor, verifying whether a polynomial is divisible by a specific factor, and solving polynomial equations.

Q: Can I use the remainder theorem to find the roots of a polynomial?

A: Yes, you can use the remainder theorem to find the roots of a polynomial. If the remainder is zero when the polynomial is divided by a linear factor, then the linear factor is a factor of the polynomial and the root of the polynomial is the value of the linear factor that makes the polynomial equal to zero.