Applying the Factor Theorem to the Problem

Oct 13, 2025 by windailys.com 30 views

If ${$ (x+k)$}$ is a factor of ${$ f(x)$}$, which of the following must be true?

Understanding the Problem

When we say that ${$ (x+k)$}$ is a factor of ${$ f(x)$}$, it means that ${$ f(x)$}$ can be expressed as a product of ${$ (x+k)$}$ and another polynomial, ${$ g(x)$}$. In other words, ${$ f(x) = (x+k) \cdot g(x)$}$. This is a fundamental concept in algebra, and it has important implications for solving equations and finding roots.

The Factor Theorem

The factor theorem states that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem is a direct consequence of the definition of a factor. If ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(x) = (x-a) \cdot g(x)$}$ for some polynomial ${$ g(x)$}$. Substituting ${$ x = a$}$ into this equation, we get ${$ f(a) = (a-a) \cdot g(a) = 0 \cdot g(a) = 0$}$.

Applying the Factor Theorem to the Problem

Now, let's apply the factor theorem to the problem at hand. We are given that ${$ (x+k)$}$ is a factor of ${$ f(x)$}$. Using the factor theorem, we can conclude that ${$ f(-k) = 0$}$. This is because ${$ (x+k)$}$ is equivalent to ${$ (x-(-k))$}$, and the factor theorem tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$.

Evaluating the Answer Choices

Now, let's evaluate the answer choices in light of our analysis.

A. ${$ f(k) = 0$}$. This is not necessarily true. The factor theorem only tells us that ${$ f(-k) = 0$}$, not ${$ f(k) = 0$}$.

B. ${$ f(-k) = 0$}$. This is true, as we have just shown using the factor theorem.

C. A root of ${$ f(x)$}$ is ${$ x = k$}$. This is not necessarily true. The factor theorem only tells us that ${$ f(-k) = 0$}$, not that ${$ x = k$}$ is a root of ${$ f(x)$}$.

D. A ${$ y$}$-intercept of ${$ f(x)$}$ is ${$ x = k$}$. This is not necessarily true. The factor theorem only tells us that ${$ f(-k) = 0$}$, not that ${$ x = k$}$ is a ${$ y$}$-intercept of ${$ f(x)$}$.

Conclusion

In conclusion, the correct answer is B. ${$ f(-k) = 0$}$. This is a direct consequence of the factor theorem, which tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. We can apply this theorem to the problem at hand to conclude that ${$ f(-k) = 0$}$.

Additional Discussion

It's worth noting that the factor theorem has many important implications for solving equations and finding roots. For example, if we know that ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$. This is because ${$ f(x) = (x-a) \cdot g(x)$}$, and substituting ${$ x = a$}$ into this equation, we get ${$ f(a) = (a-a) \cdot g(a) = 0 \cdot g(a) = 0$}$.

Example

Let's consider an example to illustrate the factor theorem. Suppose we know that ${$ (x-2)$}$ is a factor of ${$ f(x) = x^2 - 4x + 3$}$. Using the factor theorem, we can conclude that ${$ f(2) = 0$}$. This is because ${$ (x-2)$}$ is equivalent to ${$ (x-(-2))$}$, and the factor theorem tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$.

Conclusion

In conclusion, the factor theorem is a powerful tool for solving equations and finding roots. It tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. We can apply this theorem to the problem at hand to conclude that ${$ f(-k) = 0$}$.

Final Answer

The final answer is B. ${$ f(-k) = 0$}$.
Q&A: If ${$ (x+k)$}$ is a factor of ${$ f(x)$}$, which of the following must be true?

Q: What is the factor theorem?

A: The factor theorem states that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem is a direct consequence of the definition of a factor.

Q: How can we apply the factor theorem to the problem at hand?

A: We can apply the factor theorem by substituting ${$ x = -k$}$ into the equation ${$ f(x) = (x+k) \cdot g(x)$}$. This gives us ${$ f(-k) = (-k+k) \cdot g(-k) = 0 \cdot g(-k) = 0$}$.

Q: What is the relationship between the factor theorem and the roots of a polynomial?

A: The factor theorem tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ x = a$}$ is a root of ${$ f(x)$}$. This is because ${$ f(x) = (x-a) \cdot g(x)$}$, and substituting ${$ x = a$}$ into this equation, we get ${$ f(a) = (a-a) \cdot g(a) = 0 \cdot g(a) = 0$}$.

Q: Can we conclude that ${$ x = k$}$ is a root of ${$ f(x)$}$ if ${$ (x+k)$}$ is a factor of ${$ f(x)$}$?

A: No, we cannot conclude that ${$ x = k$}$ is a root of ${$ f(x)$}$ if ${$ (x+k)$}$ is a factor of ${$ f(x)$}$. The factor theorem only tells us that ${$ f(-k) = 0$}$, not that ${$ x = k$}$ is a root of ${$ f(x)$}$.

Q: What is the relationship between the factor theorem and the y-intercept of a polynomial?

A: The factor theorem does not tell us anything about the y-intercept of a polynomial. The y-intercept is the point where the graph of the polynomial intersects the y-axis, and it is not related to the factor theorem.

Q: Can we conclude that ${$ f(k) = 0$}$ if ${$ (x+k)$}$ is a factor of ${$ f(x)$}$?

A: No, we cannot conclude that ${$ f(k) = 0$}$ if ${$ (x+k)$}$ is a factor of ${$ f(x)$}$. The factor theorem only tells us that ${$ f(-k) = 0$}$, not that ${$ f(k) = 0$}$.

Q: What is the significance of the factor theorem in algebra?

A: The factor theorem is a powerful tool for solving equations and finding roots. It tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem has many important implications for solving equations and finding roots.

Q: Can we use the factor theorem to find the roots of a polynomial?

A: Yes, we can use the factor theorem to find the roots of a polynomial. If we know that ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$.

Q: What is the relationship between the factor theorem and the concept of a root?

Q: Can we use the factor theorem to find the y-intercept of a polynomial?

A: No, we cannot use the factor theorem to find the y-intercept of a polynomial. The y-intercept is the point where the graph of the polynomial intersects the y-axis, and it is not related to the factor theorem.

Q: What is the significance of the factor theorem in solving equations?

A: The factor theorem is a powerful tool for solving equations. It tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem has many important implications for solving equations.

Q: Can we use the factor theorem to find the roots of a polynomial with multiple factors?

A: Yes, we can use the factor theorem to find the roots of a polynomial with multiple factors. If we know that ${$ (x-a)$}$ and ${$ (x-b)$}$ are factors of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ and ${$ x = b$}$ are roots of ${$ f(x)$}$.

Q: What is the relationship between the factor theorem and the concept of a polynomial with multiple factors?

A: The factor theorem tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This is also true if ${$ (x-a)$}$ and ${$ (x-b)$}$ are factors of ${$ f(x)$}$. In this case, we can conclude that ${$ x = a$}$ and ${$ x = b$}$ are roots of ${$ f(x)$}$.

Q: Can we use the factor theorem to find the roots of a polynomial with complex factors?

A: Yes, we can use the factor theorem to find the roots of a polynomial with complex factors. If we know that ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$.

Q: What is the significance of the factor theorem in solving equations with complex roots?

A: The factor theorem is a powerful tool for solving equations with complex roots. It tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem has many important implications for solving equations with complex roots.

Q: Can we use the factor theorem to find the roots of a polynomial with irrational factors?

A: Yes, we can use the factor theorem to find the roots of a polynomial with irrational factors. If we know that ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$.

Q: What is the relationship between the factor theorem and the concept of a polynomial with irrational factors?

A: The factor theorem tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This is also true if ${$ (x-a)$}$ is an irrational factor of ${$ f(x)$}$. In this case, we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$.

Q: Can we use the factor theorem to find the roots of a polynomial with rational factors?

A: Yes, we can use the factor theorem to find the roots of a polynomial with rational factors. If we know that ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ is a root of ${$ f(x)$}$.

Q: What is the significance of the factor theorem in solving equations with rational roots?

A: The factor theorem is a powerful tool for solving equations with rational roots. It tells us that if ${$ (x-a)$}$ is a factor of ${$ f(x)$}$, then ${$ f(a) = 0$}$. This theorem has many important implications for solving equations with rational roots.

Q: Can we use the factor theorem to find the roots of a polynomial with mixed factors?

A: Yes, we can use the factor theorem to find the roots of a polynomial with mixed factors. If we know that ${$ (x-a)$}$ and ${$ (x-b)$}$ are factors of ${$ f(x)$}$, then we can conclude that ${$ x = a$}$ and ${$ x = b$}$ are roots of ${$ f(x)$}$.