The Polynomial Function F ( X ) = X 3 − 13 X 2 + 52 X − 60 F(x) = X^3 - 13x^2 + 52x - 60 F ( X ) = X 3 − 13 X 2 + 52 X − 60 Has X = 5 X = 5 X = 5 As A Root.1. What Are The Remaining Roots Of F ( X F(x F ( X ]? Separate Multiple Answers With A Comma.2. Enter The Factored Form Of F ( X F(x F ( X ].

Introduction to Polynomial Functions

Polynomial functions are a fundamental concept in algebra, and they play a crucial role in various mathematical and real-world applications. 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. In this article, we will focus on a specific polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$ and its roots.

The Given Polynomial Function

The given polynomial function is $f(x) = x^3 - 13x^2 + 52x - 60$. We are also given that $x = 5$ is a root of this function. This means that when we substitute $x = 5$ into the function, the result is equal to zero.

Finding the Remaining Roots

To find the remaining roots of the function, we can use the fact that if $x = 5$ is a root, then $(x - 5)$ is a factor of the function. We can use polynomial division or synthetic division to divide the function by $(x - 5)$ and find the remaining quadratic factor.

Using Synthetic Division

To use synthetic division, we need to divide the function by $(x - 5)$. We can do this by following these steps:

1. Write down the coefficients of the function in descending order of powers of $x$: $1, -13, 52, -60$.
2. Write down the root $x = 5$.
3. Multiply the root by the first coefficient and write the result below the second coefficient.
4. Add the numbers in the second column and write the result below the third coefficient.
5. Multiply the root by the result in the second column and write the result below the fourth coefficient.
6. Add the numbers in the third column and write the result below the fifth coefficient.

By following these steps, we get:

The result is a quadratic function $f(x) = (x - 5)(x^2 - 8x + 23)$.

Finding the Remaining Roots of the Quadratic Function

To find the remaining roots of the quadratic function, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -8$, and $c = 23$.

Using the Quadratic Formula

We can plug in the values of $a$, $b$, and $c$ into the quadratic formula to find the remaining roots:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(23)}}{2(1)}$ $x = \frac{8 \pm \sqrt{64 - 92}}{2}$ $x = \frac{8 \pm \sqrt{-28}}{2}$

Since the square root of a negative number is not a real number, the quadratic function has no real roots. However, we can express the roots in terms of complex numbers.

Expressing the Roots in Terms of Complex Numbers

We can express the roots in terms of complex numbers by using the fact that $\sqrt{-28} = \sqrt{28}i$, where $i$ is the imaginary unit.

$x = \frac{8 \pm \sqrt{28}i}{2}$ $x = 4 \pm \sqrt{7}i$

Factored Form of the Polynomial Function

The factored form of the polynomial function is $f(x) = (x - 5)(x - (4 + \sqrt{7}i))(x - (4 - \sqrt{7}i))$.

Conclusion

In this article, we have found the remaining roots of the polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$ and expressed the function in its factored form. We have also used synthetic division and the quadratic formula to find the remaining roots of the function. The factored form of the function is $f(x) = (x - 5)(x - (4 + \sqrt{7}i))(x - (4 - \sqrt{7}i))$.

Introduction

In our previous article, we explored the polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$ and its roots. We found that $x = 5$ is a root of the function and used synthetic division to divide the function by $(x - 5)$ and find the remaining quadratic factor. We then used the quadratic formula to find the remaining roots of the quadratic function. In this article, we will answer some frequently asked questions about the polynomial function and its roots.

Q&A

Q: What is the factored form of the polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$?

A: The factored form of the polynomial function is $f(x) = (x - 5)(x - (4 + \sqrt{7}i))(x - (4 - \sqrt{7}i))$.

Q: What are the remaining roots of the polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$?

A: The remaining roots of the polynomial function are $4 + \sqrt{7}i$ and $4 - \sqrt{7}i$.

Q: How did you find the remaining roots of the polynomial function?

A: We used synthetic division to divide the function by $(x - 5)$ and find the remaining quadratic factor. We then used the quadratic formula to find the remaining roots of the quadratic function.

Q: What is the significance of the imaginary unit $i$ in the roots of the polynomial function?

A: The imaginary unit $i$ is used to express the roots of the polynomial function in terms of complex numbers. This is because the quadratic function has no real roots, and the square root of a negative number is not a real number.

Q: Can you explain the concept of synthetic division in more detail?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It involves writing down the coefficients of the polynomial in descending order of powers of $x$, and then multiplying the root by the first coefficient and writing the result below the second coefficient. We then add the numbers in the second column and write the result below the third coefficient, and so on.

Q: What is the quadratic formula, and how is it used to find the roots of a quadratic function?

A: The quadratic formula is a formula that is used to find the roots of a quadratic function. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic function.

Q: Can you provide an example of how to use the quadratic formula to find the roots of a quadratic function?

A: Let's consider the quadratic function $f(x) = x^2 + 5x + 6$. We can use the quadratic formula to find the roots of this function by plugging in the values of $a$, $b$, and $c$ into the formula. We get:

$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$ $x = \frac{-5 \pm \sqrt{25 - 24}}{2}$ $x = \frac{-5 \pm \sqrt{1}}{2}$ $x = \frac{-5 \pm 1}{2}$

Therefore, the roots of the quadratic function are $x = -3$ and $x = -2$.

Conclusion

In this article, we have answered some frequently asked questions about the polynomial function $f(x) = x^3 - 13x^2 + 52x - 60$ and its roots. We have explained the concept of synthetic division and the quadratic formula, and provided examples of how to use these methods to find the roots of a polynomial function. We hope that this article has been helpful in clarifying any questions you may have had about the polynomial function and its roots.