If (x+8) Is A Factor Of F(x) What Must Be True A Detailed Explanation

Introduction

In the realm of polynomial functions, understanding the relationship between factors and roots is crucial. When we are given that $(x+8)$ is a factor of a polynomial function $f(x)$, it implies a specific characteristic about the roots of the function. This article will delve into the factor theorem and its implications, specifically addressing the question of what must be true if $(x+8)$ is a factor of $f(x)$. We'll explore the concepts of factors, roots, and how they connect through the factor theorem. This foundational understanding is essential for solving various problems in algebra and calculus, particularly those involving polynomial equations and their solutions. Our primary focus will be to explain the correct answer choice and elucidate why the other options are incorrect, thereby reinforcing the core principles of polynomial factorization and root identification. By the end of this discussion, you should have a firm grasp of how to apply the factor theorem in similar contexts, enhancing your problem-solving skills in mathematics.

Understanding the Factor Theorem

The factor theorem is a fundamental concept in algebra that links the factors of a polynomial to its roots. To fully grasp the implications when $(x+8)$ is a factor of $f(x)$, we must first understand this theorem. The factor theorem states that for a polynomial $f(x)$, if $f(c) = 0$ for some value $c$, then $(x-c)$ is a factor of $f(x)$. Conversely, if $(x-c)$ is a factor of $f(x)$, then $f(c) = 0$. This bidirectional relationship is key to solving problems involving polynomial roots and factors. In simpler terms, the factor theorem provides a direct way to find factors of a polynomial by identifying its roots and vice versa.

When we say $(x+8)$ is a factor of $f(x)$, it means that $f(x)$ can be written as $(x+8) imes g(x)$, where $g(x)$ is another polynomial. This implies that when $x$ is a value that makes $(x+8)$ equal to zero, $f(x)$ will also be zero. Solving the equation $x+8=0$, we find that $x=-8$. Therefore, according to the factor theorem, if $(x+8)$ is a factor of $f(x)$, then $f(-8)$ must be equal to zero. This is because substituting $x=-8$ into the factored form $(x+8) imes g(x)$ results in $(-8+8) imes g(-8) = 0 imes g(-8) = 0$. This direct application of the factor theorem highlights the crucial link between a factor of a polynomial and the corresponding root of the polynomial function. Understanding this connection is vital for tackling more complex problems in polynomial algebra.

Analyzing the Given Options

To determine the correct answer, let's analyze each option in the context of the factor theorem and the given information that $(x+8)$ is a factor of $f(x)$.

• A. Both $x=-8$ and $x=8$ are roots of $f(x)$.

  This option suggests that both $-8$ and $8$ are roots of $f(x)$. However, we know that if $(x+8)$ is a factor, only $x=-8$ corresponds to this factor. The value $x=8$ would correspond to a factor of $(x-8)$, which is not given. Therefore, this option is incorrect. To further clarify, a root of a polynomial function is a value of $x$ that makes the function equal to zero. The factor $(x+8)$ implies that setting $x=-8$ will make the polynomial zero, but setting $x=8$ does not necessarily do so unless $(x-8)$ is also a factor of $f(x)$, which we don't know to be true. Understanding this distinction is essential for correctly applying the factor theorem and avoiding common mistakes in polynomial algebra.

• B. Neither $x=-8$ nor $x=8$ is a root of $f(x)$.

  This option contradicts the factor theorem directly. If $(x+8)$ is a factor of $f(x)$, then $x=-8$ must be a root. This is because, as explained earlier, the factor theorem explicitly states that if $(x-c)$ is a factor, then $c$ is a root. In our case, $c=-8$. So, the statement that neither $-8$ nor $8$ are roots is false. It’s crucial to remember the fundamental connection between factors and roots: a factor of the form $(x-c)$ directly implies that $x=c$ is a root of the polynomial. This option incorrectly denies this relationship, making it an incorrect choice. Recognizing such direct contradictions with the factor theorem helps in quickly eliminating incorrect options in problem-solving scenarios.

• C. $f(-8)=0$

  This option aligns perfectly with the factor theorem. If $(x+8)$ is a factor of $f(x)$, then substituting $x=-8$ into $f(x)$ will result in zero. This is a direct application of the theorem, and therefore, this option is the correct answer. To reiterate, the factor theorem posits that if $(x-c)$ is a factor of $f(x)$, then $f(c) = 0$. Here, $c=-8$, so $f(-8)$ must indeed be zero. This option precisely captures the core implication of the factor theorem and correctly identifies the relationship between the given factor and the value of the function at the corresponding root. Confirming such direct applications of theorems is key to validating the correctness of an answer in mathematical problems.

• D. $f(8)=0$

  This option is incorrect because it suggests that $x=8$ is a root, which would imply that $(x-8)$ is a factor. We are only given that $(x+8)$ is a factor, not $(x-8)$. There is no information provided that would lead us to conclude that $f(8)$ equals zero. To clarify, if $f(8)$ were zero, then $(x-8)$ would have to be a factor of $f(x)$. Since we have no evidence or given information supporting this, we cannot assume that $f(8) = 0$. This option highlights the importance of sticking strictly to the information provided and not making unwarranted assumptions based on similar but distinct mathematical relationships. Accurate problem-solving requires distinguishing between what is given and what can be logically deduced from the given information.

Conclusion

In conclusion, if $(x+8)$ is a factor of $f(x)$, then the correct answer is C. $f(-8)=0$. This conclusion is a direct application of the factor theorem, which states that if $(x-c)$ is a factor of $f(x)$, then $f(c) = 0$. Understanding and applying the factor theorem is crucial for solving problems involving polynomial functions and their roots. The other options were incorrect because they either contradicted the factor theorem or made assumptions not supported by the given information. Mastering this concept provides a solid foundation for more advanced topics in algebra and calculus, enabling a deeper understanding of polynomial behavior and equation solving. By carefully analyzing the relationship between factors and roots, we can accurately determine the properties of polynomial functions and solve related problems effectively.