Using the Remainder Theorem to Find the First Root

Oct 13, 2025 by leolavanderia.com.br 108 views

Introduction

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 will use the Remainder Theorem to find all the roots of a given cubic function, given that one of the roots is already known.

The Remainder Theorem

The Remainder Theorem states that if we divide a polynomial function $f(x)$ by a linear factor $(x - a)$ , then the remainder is equal to $f(a)$ . In other words, if we substitute $a$ into the polynomial function, the result will be the remainder.

The Given Function

The given function is $f(x) = x^3 - 9x^2 + 26x - 24$ . We are told that one of the roots of this function is $x = 2$ . This means that when we substitute $x = 2$ into the function, the result will be zero.

Using the Remainder Theorem to Find the First Root

Let's use the Remainder Theorem to find the remainder when $f(x)$ is divided by $(x - 2)$ . We substitute $x = 2$ into the function:

$f(2) = (2)^3 - 9(2)^2 + 26(2) - 24$

$f(2) = 8 - 36 + 52 - 24$

$f(2) = 0$

Since the remainder is zero, we know that $(x - 2)$ is a factor of the function $f(x)$ .

Factoring the Function

Now that we know that $(x - 2)$ is a factor of the function, we can factor the function as follows:

$f(x) = (x - 2)(x^2 - 7x + 12)$

Finding the Remaining Roots

To find the remaining roots, we need to find the roots of the quadratic function $x^2 - 7x + 12$ . We can do this by factoring the quadratic function:

$x^2 - 7x + 12 = (x - 3)(x - 4)$

Finding the Roots of the Quadratic Function

Now that we have factored the quadratic function, we can find the roots by setting each factor equal to zero:

$x - 3 = 0 \Rightarrow x = 3$

$x - 4 = 0 \Rightarrow x = 4$

Conclusion

We have found all the roots of the given cubic function $f(x) = x^3 - 9x^2 + 26x - 24$ . The roots are $x = 2, x = 3, x = 4$ .

Final Answer

The final answer is: $\boxed{A}$

Introduction

In our previous article, we used the Remainder Theorem to find all the roots of a given cubic function, given that one of the roots is already known. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What is the Remainder Theorem?

A: 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. It states that if we divide a polynomial function $f(x)$ by a linear factor $(x - a)$ , then the remainder is equal to $f(a)$ .

Q: How do I use the Remainder Theorem to find the roots of a polynomial function?

A: To use the Remainder Theorem, you need to substitute the value of $a$ into the polynomial function and evaluate the result. If the result is zero, then $(x - a)$ is a factor of the function, and you can factor the function accordingly.

Q: What if the remainder is not zero?

A: If the remainder is not zero, then $(x - a)$ is not a factor of the function, and you cannot use the Remainder Theorem to find the roots.

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

A: Yes, you can use the Remainder Theorem to find the roots of a quadratic function. However, you will need to factor the quadratic function first, and then use the Remainder Theorem to find the roots of each factor.

Q: What if I have a polynomial function with multiple roots?

A: If you have a polynomial function with multiple roots, you can use the Remainder Theorem to find each root individually. You will need to substitute each root into the function and evaluate the result. If the result is zero, then you have found a root.

Q: Can I use the Remainder Theorem to find the roots of a polynomial function with complex roots?

A: Yes, you can use the Remainder Theorem to find the roots of a polynomial function with complex roots. However, you will need to use complex numbers to evaluate the result.

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

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

Not substituting the correct value of $a$ into the function
Not evaluating the result correctly
Not factoring the function correctly
Not using the correct method to find the roots

Conclusion

We hope this Q&A section has helped clarify any doubts and provided additional information on the topic of using the Remainder Theorem to find the roots of a polynomial function. Remember to always follow the steps carefully and avoid common mistakes to ensure accurate results.

Final Tips

Always read the problem carefully and understand what is being asked.
Use the Remainder Theorem to find the roots of a polynomial function, but also consider other methods, such as factoring or using the quadratic formula.
Be careful when working with complex numbers and ensure that you are using the correct method to find the roots.
Practice, practice, practice! The more you practice using the Remainder Theorem, the more comfortable you will become with the method.

Final Answer

The final answer is: $\boxed{A}$