One Root Of $f(x)=x^3-4x^2-20x+48$ Is $x=6$. What Are All The Factors Of The Function? Use The Remainder Theorem.A. $(x+6)(x+8)$B. \$(x-6)(x-8)$[/tex\]C. $(x-2)(x+4)(x-6)$D.

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Introduction

When given a polynomial function and one of its roots, we can use the Remainder Theorem to find the other factors of the function. In this article, we will explore how to apply the Remainder Theorem to factor a cubic function, given one of its roots. We will use the function $f(x)=x3-4x2-20x+48$ and the known root $x=6$ to find all the factors of the function.

The Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that relates the remainder of a polynomial function when divided by a linear factor to the value of the function at a specific point. If a polynomial function $f(x)$ is divided by $x-a$, then the remainder is equal to $f(a)$. In other words, if we know the value of the function at a specific point, we can use the Remainder Theorem to find the remainder of the function when divided by a linear factor.

Applying the Remainder Theorem to the Given Function

Given the function $f(x)=x3-4x2-20x+48$ and the known root $x=6$, we can use the Remainder Theorem to find the remainder of the function when divided by $x-6$. According to the Remainder Theorem, the remainder is equal to $f(6)$. We can substitute $x=6$ into the function to find the remainder:

f(6)=(6)3−4(6)2−20(6)+48f(6)=(6)^3-4(6)^2-20(6)+48

f(6)=216−144−120+48f(6)=216-144-120+48

f(6)=0f(6)=0

Since the remainder is equal to 0, we know that $x-6$ is a factor of the function.

Factoring the Function

Now that we know $x-6$ is a factor of the function, we can use polynomial division or synthetic division to divide the function by $x-6$ and find the other factor. Let's use polynomial division to divide the function by $x-6$:

x3−4x2−20x+48x−6\frac{x^3-4x^2-20x+48}{x-6}

=x2+2x−8=x^2+2x-8

So, the other factor of the function is $x^2+2x-8$.

Finding the Remaining Factors

To find the remaining factors of the function, we can factor the quadratic expression $x^2+2x-8$. We can use factoring by grouping or the quadratic formula to factor the expression:

x2+2x−8=(x+4)(x−2)x^2+2x-8=(x+4)(x-2)

So, the remaining factors of the function are $x+4$ and $x-2$.

Conclusion

In conclusion, we have used the Remainder Theorem to find the factors of the function $f(x)=x3-4x2-20x+48$, given the known root $x=6$. We have found that the factors of the function are $(x-6)(x+4)(x-2)$. This is the correct answer, which is option C.

Discussion

The Remainder Theorem is a powerful tool in algebra that allows us to find the remainder of a polynomial function when divided by a linear factor. In this article, we have used the Remainder Theorem to find the factors of a cubic function, given one of its roots. We have shown that the factors of the function are $(x-6)(x+4)(x-2)$, which is the correct answer.

Final Answer

The final answer is: (x−6)(x+4)(x−2)\boxed{(x-6)(x+4)(x-2)}

Introduction

In our previous article, we explored how to use the Remainder Theorem to find the factors of a cubic function, given one of its roots. We used the function $f(x)=x3-4x2-20x+48$ and the known root $x=6$ to find all the factors of the function. In this article, we will answer some common questions related to the topic.

Q&A

Q1: What is the Remainder Theorem?

A1: The Remainder Theorem is a fundamental concept in algebra that relates the remainder of a polynomial function when divided by a linear factor to the value of the function at a specific point. If a polynomial function $f(x)$ is divided by $x-a$, then the remainder is equal to $f(a)$.

Q2: How do I apply the Remainder Theorem to find the factors of a function?

A2: To apply the Remainder Theorem, you need to know the value of the function at a specific point. You can then use the Remainder Theorem to find the remainder of the function when divided by a linear factor. If the remainder is equal to 0, then the linear factor is a factor of the function.

Q3: What if I don't know the value of the function at a specific point?

A3: If you don't know the value of the function at a specific point, you can use other methods to find the factors of the function. For example, you can use polynomial division or synthetic division to divide the function by a linear factor.

Q4: Can I use the Remainder Theorem to find the roots of a function?

A4: Yes, you can use the Remainder Theorem to find the roots of a function. If the remainder is equal to 0 when the function is divided by a linear factor, then the linear factor is a root of the function.

Q5: What are some common mistakes to avoid when using the Remainder Theorem?

A5: Some common mistakes to avoid when using the Remainder Theorem include:

  • Not checking if the remainder is equal to 0 before concluding that the linear factor is a factor of the function.
  • Not using the correct value of the function at a specific point.
  • Not using the correct linear factor when dividing the function.

Q6: Can I use the Remainder Theorem to find the factors of a function with multiple roots?

A6: Yes, you can use the Remainder Theorem to find the factors of a function with multiple roots. However, you need to be careful when using the Remainder Theorem with multiple roots, as the remainder may not be equal to 0 for all roots.

Q7: What are some real-world applications of the Remainder Theorem?

A7: The Remainder Theorem has many real-world applications, including:

  • Finding the roots of a function to determine the stability of a system.
  • Finding the factors of a function to determine the behavior of a system.
  • Finding the remainder of a function to determine the error in a system.

Conclusion

In conclusion, the Remainder Theorem is a powerful tool in algebra that allows us to find the factors of a function, given one of its roots. We have answered some common questions related to the topic and provided some tips and tricks for using the Remainder Theorem.

Final Answer

The final answer is: (x−6)(x+4)(x−2)\boxed{(x-6)(x+4)(x-2)}