**A quartic function** is a type of **polynomial** with a **degree** of **four,** which means its highest **exponent** is four. It can be **expressed** in the form** $y = ax^4 + bx^3 + cx^2 + dx + e$** where **$a$, $b$, $c$, $d$,** and **$e$** are constants, and $a \neq 0$.

When considering the **real zeros** of a **quartic function**, specifically **$x = –4$** and **$x = –1$**, these values represent the **$x$-coordinates** where the **function intersects** the **$x$-axis.**

In a **quartic function** with these given zeros, I can **determine** a **formula** that will include these **points** by using them as roots in the **polynomial.**

For instance, if a **function** has roots at** $x = –4$** and **$x = –1$**, then **$(x + 4)$** and **$(x + 1)$** must be factors of this function. To ensure that these are the only real zeros, the **remaining factors** must not yield any **real zeros.**

Thus, the other two factors could be **complex** or **repeated factors** of the given real zeros to maintain a **degree** of **four.**

Stick around, and I’ll show you how a **quartic function** behaves on a graph with these **specific zeros** and how these **zeros** are **critical** in **determining** the shape and the **intersection** points of the **quartic function**‘s curve.

## Quartic Function With Only Two Real Zeros

In my exploration of **quartic functions**, which are **fourth-degree polynomials**, I’ve come across an interesting scenario.

A quartic function typically has the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ where $a$, $b$, $c$, $d$, and $e$ are **real coefficients**, and the variable $x$ represents the input to the function.

The quartic function can have up to four **real roots** (or **real zeroes**), but what about when it only has two?

Imagine I have a quartic equation that is known to have exactly two **real zeroes**: $x = -4$ and $x = -1$. To construct a quartic function with these zeroes, I know that $(x + 4)$ and $(x + 1)$ must be factors of the function. However, since I want exactly two **real roots**, these factors need to be squared to prevent the existence of additional **real zeroes**. Therefore, the equation of the function could be written as:

$$f(x) = (x + 4)^2(x + 1)^2$$

Expanding this out gives me the actual quartic function:

$$f(x) = x^4 + 10x^3 + 35x^2 + 50x + 16$$

The **leading term** is $x^4$, which confirms that the function is indeed a quartic. The **graph** of this function will touch the x-axis at $x = -4$ and $x = -1$, corresponding to the **real zeroes**, and will not cross it since these roots have even multiplicity.

As for the **range** of the function, it will be $[16, \infty)$ since the **leading term** has a positive coefficient, and the function has a minimum point where it achieves the value of 16. There will be a local maximum between the two **real roots**, which could also be the **inflection point**.

The **degree 4** nature of the function means the **graph** will exhibit a maximum of three turning points. Given the **real coefficients**, the **graph** will extend indefinitely in both the positive and negative directions along the y-axis.

Key Features | Description |
---|---|

Degree | 4 (quartic) |

Real Roots | $x = -4$, $x = -1$ (each repeated) |

Inflection Point | Possibly between $x = -4$ and $x = -1$ |

Range | $[16, \infty)$ |

## Example of a Quartic Function

In exploring the world of polynomials, I often come across an interesting type called a **quartic function**. This is a type of polynomial where the highest power of the variable, usually “x,” is four, which means it has a degree of four.

Now, let’s say I have a **quartic function** that has specifically two given **real zeros**: **x=–4** and **x=–1**. What does this imply for the function itself?

Well, for a **quartic equation** to have **real zeros** at **x=–4** and **x=–1**, these **zeros** must each appear as a factor twice since a quartic will have four total zeros (real or complex).

This gives us **two quadratic factors**: $(x+4)(x+4)$ and $(x+1)(x+1)$. To construct the entire **quartic polynomial**, I would multiply these **factors** together. That said, the resulting **equation** then becomes:

$$ (x+4)^2 \cdot (x+1)^2 = x^4 + 10x^3 + 35x^2 + 50x + 16 $$

This **equation** represents our **quartic function**. Importantly, should I desire to **solve** this **equation** for all zeros, I will notice that it naturally factors into the **quadratic factors** I began with, confirming that the only **real zeros** of the function are at **x=–4** and **x=–1**.

Understanding this example can help envision how **quartic polynomials** behave. They can curve up and down, possibly several times, since they can have up to four turning points.

If we graph our resulting function, we will note the presence of the two real zeros as intersecting x-axis points.

## Conclusion

In this **exploration** of **quartic functions**, I’ve discussed the properties of a **quartic function** with specified **real zeros** at **( x = -4 )** and **( x = -1 ). **

To construct such a **function,** we **consider** that each **zero** has to be accounted for twice because a **quartic function** is a polynomial of degree **four,** and **thus,** needs to have four **zeros—real** or **complex.**

Given the **real zeros**, I propose a **quartic function** in the form** $f(x) = (x + 4)^2(x + 1)^2 $**. When this is expanded, we would obtain a **quartic function** that satisfies the given conditions.

After expansion, the equation transforms into** $f(x) = x^4 + 10x^3 + 35x^2 + 50x + 20$**, which is a valid **representation** of the **requirements.**

Furthermore, it’s critical to **mention** that if a **quartic function** is expected to have only two **real zeros**, the remaining two zeros must be complex and occur in a **conjugate** pair.

This is a result of the **Fundamental Theorem** of **Algebra** and ensures the equation remains with real coefficients.

I hope my clarification helps understand how to construct a **quartic function** based on given **real zeros**. Remember, the complex zeros are not explicitly shown in this case but can be inferred based on the **fundamental properties** of **polynomials.**