Given $f(x) = 2x^3 + Kx - 6$, And $x - 2$ Is A Factor Of $f(x$\], What Is The Value Of $k$?Answer Attempt 1 Out Of 2:$\square$ Submit Answer

Introduction

In this article, we will explore the concept of polynomial functions and how to find the value of a coefficient in a given function. We will use the given function $f(x) = 2x^3 + kx - 6$ and the fact that $x - 2$ is a factor of $f(x)$ to solve for the value of $k$.

Understanding Polynomial Functions

A polynomial function is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. The coefficients of the polynomial function are the numbers that are multiplied by the powers of $x$. In the given function $f(x) = 2x^3 + kx - 6$, the coefficients are $2$, $k$, and $-6$.

Using the Factor Theorem

The factor theorem states that if $x - c$ is a factor of $f(x)$, then $f(c) = 0$. In this case, we are given that $x - 2$ is a factor of $f(x)$, so we can use the factor theorem to find the value of $k$.

Applying the Factor Theorem to the Given Function

We can apply the factor theorem to the given function $f(x) = 2x^3 + kx - 6$ by substituting $x = 2$ into the function. This gives us:

$f(2) = 2(2)^3 + k(2) - 6$

Simplifying the expression, we get:

$f(2) = 2(8) + 2k - 6$

$f(2) = 16 + 2k - 6$

$f(2) = 10 + 2k$

Since $x - 2$ is a factor of $f(x)$, we know that $f(2) = 0$. Therefore, we can set the expression equal to zero and solve for $k$:

$10 + 2k = 0$

Subtracting 10 from both sides, we get:

$2k = -10$

Dividing both sides by 2, we get:

$k = -5$

Conclusion

In this article, we used the factor theorem to find the value of $k$ in the given function $f(x) = 2x^3 + kx - 6$. We applied the factor theorem by substituting $x = 2$ into the function and solving for $k$. The value of $k$ is $-5$.

Final Answer

Introduction

In our previous article, we explored the concept of polynomial functions and how to find the value of a coefficient in a given function. We used the given function $f(x) = 2x^3 + kx - 6$ and the fact that $x - 2$ is a factor of $f(x)$ to solve for the value of $k$. In this article, we will answer some common questions related to the topic.

Q: What is the factor theorem?

A: The factor theorem states that if $x - c$ is a factor of $f(x)$, then $f(c) = 0$. This means that if we substitute $x = c$ into the function, the result will be zero.

Q: How do I apply the factor theorem to a given function?

A: To apply the factor theorem, we need to substitute the value of $x$ that makes $x - c$ a factor of the function into the function. This will give us an equation that we can solve for the value of the coefficient.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be written in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. A rational function, on the other hand, is a function that can be written in the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions.

Q: How do I find the value of a coefficient in a polynomial function?

A: To find the value of a coefficient in a polynomial function, we need to use the given information and the properties of the function. In the case of the factor theorem, we can use the fact that $x - c$ is a factor of the function to find the value of the coefficient.

Q: What are some common mistakes to avoid when solving for the value of a coefficient in a polynomial function?

A: Some common mistakes to avoid when solving for the value of a coefficient in a polynomial function include:

• Not using the correct value of $x$ when applying the factor theorem
• Not simplifying the expression correctly
• Not solving for the correct value of the coefficient

Q: How do I check my answer when solving for the value of a coefficient in a polynomial function?

A: To check your answer when solving for the value of a coefficient in a polynomial function, you can substitute the value of $x$ back into the function and verify that the result is zero. You can also use a calculator or computer software to check your answer.

Conclusion

In this article, we answered some common questions related to solving for the value of a coefficient in a polynomial function. We covered topics such as the factor theorem, applying the factor theorem, and common mistakes to avoid. We also provided some tips for checking your answer.

Final Answer

The final answer is $\boxed{-5}$.