The Polynomial Function Q ( X ) = X 4 + 3 X 3 − 6 X 2 − 6 X + 8 Q(x) = X^4 + 3x^3 - 6x^2 - 6x + 8 Q ( X ) = X 4 + 3 X 3 − 6 X 2 − 6 X + 8 Has X = 2 X = \sqrt{2} X = 2 ​ As A Root.1. What Are The Remaining Roots Of Q ( X Q(x Q ( X ]? Separate Multiple Answers With A Comma.2. Enter The Factored Form Of Q ( X Q(x Q ( X ].

Introduction

In mathematics, polynomial functions are a fundamental concept in algebra, and understanding their roots is crucial for solving various mathematical problems. A polynomial function is a function of 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 the polynomial function $q(x) = x^4 + 3x^3 - 6x^2 - 6x + 8$ and its roots.

The Given Root

We are given that $x = \sqrt{2}$ is a root of the polynomial function $q(x)$. This means that when we substitute $x = \sqrt{2}$ into the function, the result is equal to zero. Mathematically, this can be expressed as:

$$q(\sqrt{2}) = (\sqrt{2})^4 + 3(\sqrt{2})^3 - 6(\sqrt{2})^2 - 6(\sqrt{2}) + 8 = 0$$

Finding the Remaining Roots

To find the remaining roots of the polynomial function $q(x)$, we can use the fact that if $x = a$ is a root of the function, then $(x - a)$ is a factor of the function. Since we know that $x = \sqrt{2}$ is a root, we can write:

$$q(x) = (x - \sqrt{2}) \cdot p(x)$$

where $p(x)$ is a polynomial function of degree 3. To find the remaining roots, we need to find the roots of the polynomial function $p(x)$.

Dividing the Polynomial

To find the polynomial function $p(x)$, we can divide the polynomial function $q(x)$ by $(x - \sqrt{2})$. This can be done using long division or synthetic division. After performing the division, we get:

$$q(x) = (x - \sqrt{2})(x^3 + 2\sqrt{2}x^2 + 3x + 2\sqrt{2} + 4)$$

Finding the Roots of the Cubic Polynomial

Now, we need to find the roots of the cubic polynomial $x^3 + 2\sqrt{2}x^2 + 3x + 2\sqrt{2} + 4$. To do this, we can use various methods such as factoring, the Rational Root Theorem, or numerical methods. After analyzing the polynomial, we find that it can be factored as:

$$x^3 + 2\sqrt{2}x^2 + 3x + 2\sqrt{2} + 4 = (x + \sqrt{2} + 2)(x^2 + x + 2)$$

Finding the Roots of the Quadratic Polynomial

Now, we need to find the roots of the quadratic polynomial $x^2 + x + 2$. To do this, we can use the quadratic formula:

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

where $a = 1$, $b = 1$, and $c = 2$. Plugging in these values, we get:

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

Simplifying the expression, we get:

$$x = \frac{-1 \pm \sqrt{-7}}{2}$$

Since the discriminant is negative, the quadratic polynomial has no real roots.

The Remaining Roots

Now, we can find the remaining roots of the polynomial function $q(x)$. We have already found one root, $x = \sqrt{2}$. The other roots are the roots of the cubic polynomial $x^3 + 2\sqrt{2}x^2 + 3x + 2\sqrt{2} + 4$. We have factored this polynomial as:

$$(x + \sqrt{2} + 2)(x^2 + x + 2)$$

The roots of the quadratic polynomial $x^2 + x + 2$ are complex numbers, and the root of the linear polynomial $x + \sqrt{2} + 2$ is a real number. Therefore, the remaining roots of the polynomial function $q(x)$ are:

$$x = -\sqrt{2} - 2, x = \frac{-1 + \sqrt{-7}}{2}, x = \frac{-1 - \sqrt{-7}}{2}$$

The Factored Form of the Polynomial

Now, we can write the factored form of the polynomial function $q(x)$:

$$q(x) = (x - \sqrt{2})(x + \sqrt{2} + 2)(x^2 + x + 2)$$

This is the factored form of the polynomial function $q(x)$.

Conclusion

In this article, we have found the remaining roots of the polynomial function $q(x) = x^4 + 3x^3 - 6x^2 - 6x + 8$ and its factored form. We have used the fact that if $x = a$ is a root of the function, then $(x - a)$ is a factor of the function. We have also used various methods such as factoring, the Rational Root Theorem, and numerical methods to find the roots of the polynomial function. The remaining roots of the polynomial function $q(x)$ are:

$$x = -\sqrt{2} - 2, x = \frac{-1 + \sqrt{-7}}{2}, x = \frac{-1 - \sqrt{-7}}{2}$$

The factored form of the polynomial function $q(x)$ is:

$$q(x) = (x - \sqrt{2})(x + \sqrt{2} + 2)(x^2 + x + 2)$$

Introduction

In our previous article, we explored the polynomial function $q(x) = x^4 + 3x^3 - 6x^2 - 6x + 8$ and its roots. We found that $x = \sqrt{2}$ is a root of the function and used this information to find the remaining roots and the factored form of the polynomial. In this article, we will answer some common questions related to the polynomial function and its roots.

Q: What is the significance of the given root $x = \sqrt{2}$?

A: The given root $x = \sqrt{2}$ is significant because it allows us to factor the polynomial function $q(x)$ and find the remaining roots. By using the fact that if $x = a$ is a root of the function, then $(x - a)$ is a factor of the function, we can write the polynomial function as $(x - \sqrt{2}) \cdot p(x)$, where $p(x)$ is a polynomial function of degree 3.

Q: How do we find the remaining roots of the polynomial function?

A: To find the remaining roots of the polynomial function, we need to find the roots of the polynomial function $p(x)$. We can do this by factoring the polynomial function or using numerical methods such as the Rational Root Theorem or the quadratic formula.

Q: What is the factored form of the polynomial function $q(x)$?

A: The factored form of the polynomial function $q(x)$ is:

$$q(x) = (x - \sqrt{2})(x + \sqrt{2} + 2)(x^2 + x + 2)$$

Q: What are the remaining roots of the polynomial function $q(x)$?

A: The remaining roots of the polynomial function $q(x)$ are:

$$x = -\sqrt{2} - 2, x = \frac{-1 + \sqrt{-7}}{2}, x = \frac{-1 - \sqrt{-7}}{2}$$

Q: How do we find the roots of a polynomial function?

A: There are several methods to find the roots of a polynomial function, including:

• Factoring the polynomial function
• Using the Rational Root Theorem
• Using the quadratic formula
• Using numerical methods such as Newton's method or the bisection method

Q: What is the importance of finding the roots of a polynomial function?

A: Finding the roots of a polynomial function is important because it allows us to understand the behavior of the function and make predictions about its values. The roots of a polynomial function are also used in various applications such as engineering, physics, and economics.

Q: Can we find the roots of a polynomial function using a calculator or computer?

A: Yes, we can find the roots of a polynomial function using a calculator or computer. Many calculators and computer software programs have built-in functions to find the roots of a polynomial function.

Q: What are some common mistakes to avoid when finding the roots of a polynomial function?

A: Some common mistakes to avoid when finding the roots of a polynomial function include:

• Not checking the degree of the polynomial function
• Not using the correct method to find the roots
• Not checking for complex roots
• Not using a calculator or computer to check the roots

Conclusion

In this article, we have answered some common questions related to the polynomial function and its roots. We have discussed the significance of the given root $x = \sqrt{2}$, how to find the remaining roots of the polynomial function, and the factored form of the polynomial function. We have also discussed the importance of finding the roots of a polynomial function and some common mistakes to avoid when finding the roots.