If $(x+8)$ Is A Factor Of $f(x)$, Which Of The Following Must Be True?A. Both \$x=-8$[/tex\] And $x=8$ Are Roots Of $f(x)$.B. Neither \$x=-8$[/tex\] Nor $x=8$ Is A Root

Feb 28, 2025 by ADMIN 181 views

If (x+8) is a Factor of f(x), Which of the Following Must be True?

Understanding the Factor Theorem

The factor theorem is a fundamental concept in algebra that states if a polynomial f(x) is divisible by (x - a), then f(a) = 0. In other words, if (x - a) is a factor of f(x), then a is a root of the polynomial. This theorem can be extended to other types of factors, including (x + a) and (x - a + b).

The Given Problem

In this problem, we are given that (x + 8) is a factor of f(x). We need to determine which of the following statements must be true:

A. Both x = -8 and x = 8 are roots of f(x). B. Neither x = -8 nor x = 8 is a root of f(x).

Analyzing the Factor (x + 8)

To understand the implications of (x + 8) being a factor of f(x), let's analyze the expression (x + 8). This expression is a linear factor, which means it has a single root. The root of (x + 8) can be found by setting the expression equal to zero and solving for x.

Finding the Root of (x + 8)

To find the root of (x + 8), we set the expression equal to zero:

x + 8 = 0

Subtracting 8 from both sides gives us:

x = -8

Therefore, the root of (x + 8) is x = -8.

Implications of (x + 8) Being a Factor of f(x)

Since (x + 8) is a factor of f(x), we know that x = -8 is a root of f(x). This is a direct consequence of the factor theorem. However, we need to determine whether x = 8 is also a root of f(x).

Analyzing Statement A

Statement A claims that both x = -8 and x = 8 are roots of f(x). However, we have already established that the root of (x + 8) is x = -8. There is no evidence to suggest that x = 8 is also a root of f(x). In fact, the factor (x + 8) does not provide any information about the root x = 8.

Analyzing Statement B

Statement B claims that neither x = -8 nor x = 8 is a root of f(x). However, we have already established that x = -8 is a root of f(x). Therefore, statement B is incorrect.

Conclusion

Based on our analysis, we can conclude that:

(x + 8) is a factor of f(x) implies that x = -8 is a root of f(x).
There is no evidence to suggest that x = 8 is a root of f(x).
Therefore, statement A is incorrect, and statement B is also incorrect.

However, we can rephrase statement B to make it correct:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x), and there is no evidence to suggest that x = 8 is not a root of f(x).

Final Answer

The final answer is that statement B is incorrect, and statement A is also incorrect. However, we can rephrase statement B to make it correct:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x), and there is no evidence to suggest that x = 8 is not a root of f(x).

Understanding the Factor Theorem

The Given Problem

In this problem, we are given that (x + 8) is a factor of f(x). We need to determine which of the following statements must be true:

A. Both x = -8 and x = 8 are roots of f(x). B. Neither x = -8 nor x = 8 is a root of f(x).

Analyzing the Factor (x + 8)

Finding the Root of (x + 8)

To find the root of (x + 8), we set the expression equal to zero:

x + 8 = 0

Subtracting 8 from both sides gives us:

x = -8

Therefore, the root of (x + 8) is x = -8.

Implications of (x + 8) Being a Factor of f(x)

Since (x + 8) is a factor of f(x), we know that x = -8 is a root of f(x). This is a direct consequence of the factor theorem. However, we need to determine whether x = 8 is also a root of f(x).

Analyzing Statement A

Analyzing Statement B

Statement B claims that neither x = -8 nor x = 8 is a root of f(x). However, we have already established that x = -8 is a root of f(x). Therefore, statement B is incorrect.

Conclusion

Based on our analysis, we can conclude that:

(x + 8) is a factor of f(x) implies that x = -8 is a root of f(x).
There is no evidence to suggest that x = 8 is a root of f(x).
Therefore, statement A is incorrect, and statement B is also incorrect.

However, we can rephrase statement B to make it correct:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x), and there is no evidence to suggest that x = 8 is not a root of f(x).

Final Answer

The final answer is that statement B is incorrect, and statement A is also incorrect. However, we can rephrase statement B to make it correct:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x), and there is no evidence to suggest that x = 8 is not a root of f(x).
Q&A: Understanding the Factor Theorem and Its Implications

Q: What is the factor theorem?

A: The factor theorem is a fundamental concept in algebra that states if a polynomial f(x) is divisible by (x - a), then f(a) = 0. In other words, if (x - a) is a factor of f(x), then a is a root of the polynomial.

Q: How does the factor theorem relate to the given problem?

A: In the given problem, we are told that (x + 8) is a factor of f(x). Using the factor theorem, we can conclude that x = -8 is a root of f(x).

Q: What is the significance of the root x = -8?

A: The root x = -8 is significant because it is a direct consequence of the factor theorem. Since (x + 8) is a factor of f(x), we know that x = -8 is a root of f(x).

Q: What about the root x = 8?

A: There is no evidence to suggest that x = 8 is a root of f(x). The factor (x + 8) does not provide any information about the root x = 8.

Q: Can we conclude that x = 8 is not a root of f(x)?

A: No, we cannot conclude that x = 8 is not a root of f(x. We can only say that there is no evidence to suggest that x = 8 is a root of f(x.

Q: How can we rephrase statement B to make it correct?

A: We can rephrase statement B to make it correct by changing it to:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x, and there is no evidence to suggest that x = 8 is not a root of f(x.

Q: What is the final answer to the given problem?

A: The final answer is that statement B is incorrect, and statement A is also incorrect. However, we can rephrase statement B to make it correct:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x, and there is no evidence to suggest that x = 8 is not a root of f(x.

Frequently Asked Questions

Q: What is the factor theorem? A: The factor theorem is a fundamental concept in algebra that states if a polynomial f(x) is divisible by (x - a), then f(a) = 0. In other words, if (x - a) is a factor of f(x), then a is a root of the polynomial.
Q: How does the factor theorem relate to the given problem? A: In the given problem, we are told that (x + 8) is a factor of f(x). Using the factor theorem, we can conclude that x = -8 is a root of f(x).
Q: What is the significance of the root x = -8? A: The root x = -8 is significant because it is a direct consequence of the factor theorem. Since (x + 8) is a factor of f(x), we know that x = -8 is a root of f(x).
Q: What about the root x = 8? A: There is no evidence to suggest that x = 8 is a root of f(x). The factor (x + 8) does not provide any information about the root x = 8.
Q: Can we conclude that x = 8 is not a root of f(x)? A: No, we cannot conclude that x = 8 is not a root of f(x. We can only say that there is no evidence to suggest that x = 8 is a root of f(x.
Q: How can we rephrase statement B to make it correct? A: We can rephrase statement B to make it correct by changing it to:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x, and there is no evidence to suggest that x = 8 is not a root of f(x.

Conclusion

In conclusion, the factor theorem is a fundamental concept in algebra that states if a polynomial f(x) is divisible by (x - a), then f(a) = 0. In the given problem, we are told that (x + 8) is a factor of f(x). Using the factor theorem, we can conclude that x = -8 is a root of f(x. However, there is no evidence to suggest that x = 8 is a root of f(x. We can rephrase statement B to make it correct by changing it to:

B. Either x = -8 or x = 8 is a root of f(x).

This revised statement is consistent with our analysis, as we have established that x = -8 is a root of f(x, and there is no evidence to suggest that x = 8 is not a root of f(x.