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

Introduction

In algebra, finding the roots of a polynomial function is a crucial step in understanding its behavior and properties. Given a cubic function, we can use various methods to determine its roots. In this article, we will explore how to find all the roots of a cubic function when one factor is known.

The Given Function

The given cubic function is:

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

We are told that one factor of this function is $(x-2)$. Our goal is to find all the roots of the function.

Using the Factor Theorem

The factor theorem states that if $f(a) = 0$, then $(x-a)$ is a factor of $f(x)$. In this case, we know that $(x-2)$ is a factor of $f(x)$. This means that when we substitute $x=2$ into the function, we should get a result of zero.

Let's verify this using the given function:

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

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

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

$$f(2) = 0$$

This confirms that $(x-2)$ is indeed a factor of $f(x)$.

Dividing the Function

To find the other factors of the function, we can divide $f(x)$ by $(x-2)$. This will give us a quadratic function, which we can then factor or use the quadratic formula to find its roots.

Using polynomial long division or synthetic division, we can divide $f(x)$ by $(x-2)$:

$$\frac{4x^3 - 4x^2 - 16x + 16}{x-2} = 4x^2 + 4x - 8$$

This quadratic function has two roots, which we can find using the quadratic formula:

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

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

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

$$x = \frac{-4 \pm \sqrt{16 + 128}}{8}$$

$$x = \frac{-4 \pm \sqrt{144}}{8}$$

$$x = \frac{-4 \pm 12}{8}$$

This gives us two possible values for $x$:

$$x = \frac{-4 + 12}{8} = \frac{8}{8} = 1$$

$$x = \frac{-4 - 12}{8} = \frac{-16}{8} = -2$$

Finding the Roots

We have now found all the roots of the function:

• The factor $(x-2)$ gives us the root $x=2$.
• The quadratic function $4x^2 + 4x - 8$ has roots $x=1$ and $x=-2$.

Therefore, the roots of the function are $x=1$, $x=2$, and $x=-2$.

Conclusion

In this article, we used the factor theorem and polynomial division to find all the roots of a given cubic function. We verified that $(x-2)$ is a factor of the function and then divided the function by this factor to obtain a quadratic function. We then used the quadratic formula to find the roots of this quadratic function. The final answer is that the roots of the function are $x=1$, $x=2$, and $x=-2$.

Answer

The correct answer is:

• A. $x=1, x=2$, or $x=-2$
  One Factor of a Cubic Function: Finding All Roots - Q&A =====================================================

Introduction

In our previous article, we explored how to find all the roots of a cubic function when one factor is known. We used the factor theorem and polynomial division to determine the roots of the function. In this article, we will answer some common questions related to finding the roots of a cubic function.

Q: What is the factor theorem?

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

Q: How do I use the factor theorem to find a factor of a function?

A: To use the factor theorem, we need to substitute a value of $x$ into the function and check if the result is zero. If it is, then we have found a factor of the function. For example, if we have the function $f(x) = x^2 + 4x + 4$ and we substitute $x=2$, we get:

$$f(2) = (2)^2 + 4(2) + 4$$

$$f(2) = 4 + 8 + 4$$

$$f(2) = 16$$

Since $f(2) \neq 0$, we know that $(x-2)$ is not a factor of the function. However, if we substitute $x=-2$, we get:

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

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

$$f(-2) = 0$$

Since $f(-2) = 0$, we know that $(x+2)$ is a factor of the function.

Q: How do I divide a polynomial by a factor?

A: To divide a polynomial by a factor, we can use polynomial long division or synthetic division. These methods involve dividing the polynomial by the factor and finding the remainder. The remainder will be a polynomial of lower degree than the original polynomial.

For example, let's say we want to divide the polynomial $f(x) = x^3 + 2x^2 + 3x + 1$ by the factor $(x+1)$. We can use polynomial long division to find the quotient and remainder:

$$\frac{x^3 + 2x^2 + 3x + 1}{x+1} = x^2 + x + 1$$

The remainder is zero, which means that $(x+1)$ is a factor of the polynomial.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the roots of a quadratic equation. The quadratic equation is in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The quadratic formula is:

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

This formula can be used to find the roots of a quadratic equation.

Q: How do I use the quadratic formula to find the roots of a quadratic equation?

A: To use the quadratic formula, we need to plug in the values of $a$, $b$, and $c$ into the formula. For example, let's say we have the quadratic equation $x^2 + 4x + 4 = 0$. We can plug in the values $a=1$, $b=4$, and $c=4$ into the quadratic formula:

$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}$$

$$x = \frac{-4 \pm \sqrt{16 - 16}}{2}$$

$$x = \frac{-4 \pm \sqrt{0}}{2}$$

$$x = \frac{-4}{2}$$

$$x = -2$$

Since the discriminant is zero, we know that the quadratic equation has only one root, which is $x=-2$.

Conclusion

In this article, we answered some common questions related to finding the roots of a cubic function. We discussed the factor theorem, polynomial division, and the quadratic formula. We also provided examples of how to use these concepts to find the roots of a cubic function.