Find The Real And Complex Zeros Of The Function${ F(x) = X^4 + 6x^3 + 10x^2 + 6x + 9 }$Express Your Answer As A List Separated By Commas. Do Not Use A Plus-minus Sign { (\pm)$}$ In Your Answer; Instead, List The Zeros Separately.

Introduction

In algebra, finding the zeros of a polynomial function is a crucial step in understanding its behavior and properties. A zero of a function is a value of the variable that makes the function equal to zero. In this article, we will focus on finding the real and complex zeros of the quartic function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$. We will use various mathematical techniques, including factorization and the quadratic formula, to find the zeros of this function.

Understanding the Function

Before we begin finding the zeros of the function, let's take a closer look at its structure. The function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$ is a quartic function, meaning it has a degree of 4. This means that the highest power of the variable $x$ is 4. The coefficients of the function are all positive, which suggests that the function may have a positive leading coefficient.

Factoring the Function

One way to find the zeros of a function is to factor it into simpler polynomials. If we can factor the function, we may be able to find the zeros more easily. Let's try to factor the function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$.

After some trial and error, we can factor the function as follows:

$f(x) = (x^2 + 3x + 3)^2$

This means that the function can be written as the square of a quadratic function. We can now focus on finding the zeros of the quadratic function $x^2 + 3x + 3$.

Finding the Zeros of the Quadratic Function

To find the zeros of the quadratic function $x^2 + 3x + 3$, we can use the quadratic formula. The quadratic formula states that the zeros of a quadratic function $ax^2 + bx + c$ are given by:

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

In this case, the coefficients of the quadratic function are $a = 1$, $b = 3$, and $c = 3$. Plugging these values into the quadratic formula, we get:

$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(3)}}{2(1)}$

Simplifying the expression under the square root, we get:

$x = \frac{-3 \pm \sqrt{9 - 12}}{2}$

$x = \frac{-3 \pm \sqrt{-3}}{2}$

This means that the zeros of the quadratic function $x^2 + 3x + 3$ are complex numbers.

Finding the Zeros of the Original Function

Now that we have found the zeros of the quadratic function $x^2 + 3x + 3$, we can find the zeros of the original function $f(x) = (x^2 + 3x + 3)^2$. Since the original function is the square of the quadratic function, its zeros will be the same as the zeros of the quadratic function.

Therefore, the zeros of the original function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$ are:

$x = \frac{-3 + \sqrt{-3}}{2}$

$x = \frac{-3 - \sqrt{-3}}{2}$

$x = \frac{-3 + \sqrt{-3}}{2}$

$x = \frac{-3 - \sqrt{-3}}{2}$

Conclusion

In this article, we have found the real and complex zeros of the quartic function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$. We used various mathematical techniques, including factorization and the quadratic formula, to find the zeros of this function. The zeros of the function are complex numbers, which are given by:

$x = \frac{-3 + \sqrt{-3}}{2}$

$x = \frac{-3 - \sqrt{-3}}{2}$

$x = \frac{-3 + \sqrt{-3}}{2}$

$x = \frac{-3 - \sqrt{-3}}{2}$

Introduction

In our previous article, we discussed how to find the real and complex zeros of a quartic function. We used various mathematical techniques, including factorization and the quadratic formula, to find the zeros of the function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$. In this article, we will answer some common questions related to finding the real and complex zeros of a quartic function.

Q: What is a quartic function?

A quartic function is a polynomial function of degree 4, meaning that the highest power of the variable $x$ is 4. Quartic functions can be written in the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are constants.

Q: How do I find the zeros of a quartic function?

There are several methods to find the zeros of a quartic function, including:

• Factorization: If the quartic function can be factored into simpler polynomials, we can find the zeros by finding the zeros of the individual factors.
• Quadratic formula: If the quartic function can be written as the product of two quadratic functions, we can use the quadratic formula to find the zeros.
• Numerical methods: If the quartic function cannot be factored or written as the product of two quadratic functions, we can use numerical methods, such as the Newton-Raphson method, to find the zeros.

Q: What are the real and complex zeros of a quartic function?

The real zeros of a quartic function are the values of $x$ that make the function equal to zero. The complex zeros of a quartic function are the values of $x$ that make the function equal to zero, but are not real numbers.

Q: How do I determine if a quartic function has real or complex zeros?

To determine if a quartic function has real or complex zeros, we can use the discriminant, which is a value that can be calculated from the coefficients of the quartic function. If the discriminant is positive, the quartic function has two real zeros and two complex zeros. If the discriminant is negative, the quartic function has four complex zeros.

Q: Can a quartic function have only real zeros?

Yes, a quartic function can have only real zeros. For example, the quartic function $f(x) = x^4 - 6x^2 + 9$ has only real zeros.

Q: Can a quartic function have only complex zeros?

Yes, a quartic function can have only complex zeros. For example, the quartic function $f(x) = x^4 + 6x^3 + 10x^2 + 6x + 9$ has only complex zeros.

Q: How do I find the zeros of a quartic function with complex coefficients?

To find the zeros of a quartic function with complex coefficients, we can use the same methods as for a quartic function with real coefficients, but we need to be careful when dealing with complex numbers.

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

Some common mistakes to avoid when finding the zeros of a quartic function include:

• Not checking if the function can be factored or written as the product of two quadratic functions.
• Not using the correct method for finding the zeros, such as the quadratic formula or numerical methods.
• Not checking if the zeros are real or complex.
• Not being careful when dealing with complex numbers.

Conclusion

In this article, we have answered some common questions related to finding the real and complex zeros of a quartic function. We hope that this article has provided a clear and concise explanation of how to find the zeros of a quartic function and has helped to avoid some common mistakes.