One Factor Of $f(x) = 4x^3 - 4x^2 - 16x + 16$ Is $(x-2)$. What Are All The Roots Of The Function? Use The Remainder Theorem.A. $x = 1, X = 2$, Or $x = 4$B. $x = -2, X = 1$, Or $x = 2$C. $x =

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 the roots of a given cubic function, given that one of its factors is 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 know that $(x - a)$ is a factor of $f(x)$, then we can use the Remainder Theorem to find the value of $f(a)$.

The Given Function

The given function is $f(x) = 4x^3 - 4x^2 - 16x + 16$. We are told that one of its factors is $(x - 2)$. This means that we can use the Remainder Theorem to find the value of $f(2)$.

Finding the Value of $f(2)$

To find the value of $f(2)$, we simply substitute $x = 2$ into the function:

$f(2) = 4(2)^3 - 4(2)^2 - 16(2) + 16$

$f(2) = 4(8) - 4(4) - 32 + 16$

$f(2) = 32 - 16 - 32 + 16$

$f(2) = 0$

Conclusion

Since $f(2) = 0$, we know that $(x - 2)$ is indeed a factor of $f(x)$. This means that $x = 2$ is one of the roots of the function.

Finding the Other Roots

To find the other roots, we need to factorize the function $f(x)$ completely. We can do this by dividing the function by the known factor $(x - 2)$.

Factorizing the Function

To factorize the function, we can use polynomial long division or synthetic division. Let's use polynomial long division:

$f(x) = 4x^3 - 4x^2 - 16x + 16$

$(x - 2)$

$\underline{4x^3 - 8x^2}$

$4x^2 - 16x + 16$

$\underline{4x^2 - 8x}$

$-8x + 16$

$\underline{-8x + 16}$

$0$

The Factorized Form

The factorized form of the function is:

$f(x) = (x - 2)(4x^2 - 8x + 8)$

Finding the Other Roots

To find the other roots, we need to solve the quadratic equation $4x^2 - 8x + 8 = 0$. We can use the quadratic formula to solve this equation:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 4$, $b = -8$, and $c = 8$. Plugging these values into the formula, we get:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(8)}}{2(4)}$

$x = \frac{8 \pm \sqrt{64 - 128}}{8}$

$x = \frac{8 \pm \sqrt{-64}}{8}$

$x = \frac{8 \pm 8i}{8}$

$x = 1 \pm i$

Conclusion

The roots of the function $f(x) = 4x^3 - 4x^2 - 16x + 16$ are $x = 2$, $x = 1 + i$, and $x = 1 - i$.

Final Answer

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

Introduction

In our previous article, we used the Remainder Theorem to find the roots of a given cubic function, given that one of its factors is known. In this article, we will answer some frequently asked questions related to 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 function?

A: To use the Remainder Theorem to find the roots of a function, you need to know one of its factors. You can then substitute the value of $x$ that makes the factor equal to zero into the function to find the remainder. If the remainder is zero, then the factor is indeed a root of the function.

Q: What if I don't know any of the factors of the function?

A: If you don't know any of the factors of the function, you can try to factorize the function completely using polynomial long division or synthetic division. Once you have the factorized form of the function, you can 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 need to know one of its factors, which is a linear factor. If you know the factor, you can substitute the value of $x$ that makes the factor equal to zero into the function to find the remainder. If the remainder is zero, then the factor is indeed a root of the function.

Q: What if I get a complex root when using the Remainder Theorem?

A: If you get a complex root when using the Remainder Theorem, it means that the root is not a real number. Complex roots come in conjugate pairs, which means that if you get a complex root $a + bi$, then you will also get a complex root $a - bi$.

Q: Can I use the Remainder Theorem to find the roots of a polynomial function of degree higher than 3?

A: Yes, you can use the Remainder Theorem to find the roots of a polynomial function of degree higher than 3. However, you need to know one of its factors, which is a linear factor. If you know the factor, you can substitute the value of $x$ that makes the factor equal to zero into the function to find the remainder. If the remainder is zero, then the factor is indeed a root of the function.

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 knowing one of the factors of the function
• Not substituting the correct value of $x$ into the function
• Not checking if the remainder is zero before concluding that the factor is a root of the function
• Not considering complex roots

Conclusion

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. By using the Remainder Theorem, we can find the roots of a function given that one of its factors is known. We hope that this Q&A article has helped to clarify any doubts you may have had about the topic.

Final Answer

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