Understanding the Concept of Roots and Factors

Oct 13, 2025 by magicalceylontours.com 149 views

If ${-1}$ is a Root of ${f(x)}$ , Which of the Following Must be True?

Understanding the Concept of Roots and Factors

In mathematics, a root of a polynomial function ${f(x)}$ is a value of ${x}$ that makes the function equal to zero. In other words, if ${f(x) = 0}$ , then ${x}$ is a root of the function. On the other hand, a factor of a polynomial function is an expression that divides the function without leaving a remainder. In this article, we will explore the relationship between roots and factors of a polynomial function.

The Relationship Between Roots and Factors

If ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. This is because if ${-1}$ is a root of ${f(x)}$ , then ${f(-1) = 0}$ , and we can factor out the term ${(x + 1)}$ from ${f(x)}$ . Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ .

Must be True: A Factor of ${f(x)}$ is ${(x+1)}$

Based on the relationship between roots and factors, we can conclude that if ${-1}$ is a root of ${f(x)}$ , then a factor of ${f(x)}$ is ${(x+1)}$ . This is because if ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ .

Must be True: A Factor of ${f(x)}$ is ${(x-1)}$

However, we cannot conclude that a factor of ${f(x)}$ is ${(x-1)}$ . This is because if ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ , but we cannot conclude that ${(x-1)}$ is a factor of ${f(x)}$ .

Must be True: Both ${(x-1)}$ and ${(x+1)}$ are Factors of ${f(x)}$

We cannot conclude that both ${(x-1)}$ and ${(x+1)}$ are factors of ${f(x)}$ . This is because if ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ , but we cannot conclude that ${(x-1)}$ is a factor of ${f(x)}$ .

Conclusion

In conclusion, if ${-1}$ is a root of ${f(x)}$ , then a factor of ${f(x)}$ is ${(x+1)}$ . However, we cannot conclude that a factor of ${f(x)}$ is ${(x-1)}$ or that both ${(x-1)}$ and ${(x+1)}$ are factors of ${f(x)}$ . Therefore, the correct answer is:

A factor of ${f(x)}$ is ${(x+1)}$ .

Final Answer

The final answer is B.
If ${-1}$ is a Root of ${f(x)}$ , Which of the Following Must be True? - Q&A

Understanding the Concept of Roots and Factors

Q&A

Q: What is a root of a polynomial function?

A: A root of a polynomial function ${f(x)}$ is a value of ${x}$ that makes the function equal to zero. In other words, if ${f(x) = 0}$ , then ${x}$ is a root of the function.

Q: What is a factor of a polynomial function?

A: A factor of a polynomial function is an expression that divides the function without leaving a remainder.

Q: If ${-1}$ is a root of ${f(x)}$ , what can we conclude about the factors of ${f(x)}$ ?

A: If ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ .

Q: Must be True: A Factor of ${f(x)}$ is ${(x-1)}$ ?

A: No, we cannot conclude that a factor of ${f(x)}$ is ${(x-1)}$ . This is because if ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ , but we cannot conclude that ${(x-1)}$ is a factor of ${f(x)}$ .

Q: Must be True: Both ${(x-1)}$ and ${(x+1)}$ are Factors of ${f(x)}$ ?

A: No, we cannot conclude that both ${(x-1)}$ and ${(x+1)}$ are factors of ${f(x)}$ . This is because if ${-1}$ is a root of ${f(x)}$ , then we can write ${f(x) = (x + 1) \cdot g(x)}$ , where ${g(x)}$ is another polynomial function. Therefore, we can conclude that ${(x + 1)}$ is a factor of ${f(x)}$ , but we cannot conclude that ${(x-1)}$ is a factor of ${f(x)}$ .

Q: What is the correct answer?

A: The correct answer is B. A factor of ${f(x)}$ is ${(x+1)}$ .

Conclusion

A factor of ${f(x)}$ is ${(x+1)}$ .

Final Answer

The final answer is B.

Additional Tips and Resources

To learn more about roots and factors of polynomial functions, check out the following resources:

Khan Academy: Roots and Factors of Polynomial Functions
Mathway: Roots and Factors of Polynomial Functions
Wolfram Alpha: Roots and Factors of Polynomial Functions

Common Mistakes to Avoid

Make sure to understand the concept of roots and factors of polynomial functions before attempting to solve problems.
Be careful when writing expressions for roots and factors, as small mistakes can lead to incorrect conclusions.
Practice solving problems involving roots and factors to build your skills and confidence.