This question aims to find the **polynomial** with a **degree** **4** and given **zeros** of **-4, 3, 0, and -2**.

The question depends on the concepts of **polynomial expressions** and the **degree** of **polynomials** with **zeros.** The degree of any polynomial is the **highest exponent** of its **independent variable.** The **zeros** of a **polynomial** are the values where the **output** of the polynomial becomes **zero.**

## Expert Answer

If **c** is the **zero** of the **polynomial,** then **(x-c)** is a **factor** of the **polynomial** if and only if the polynomial is **zero** at** c**. Let the polynomial we need to find is **P(x)**. Then **-4, 3, 0, and -2** will be the **zeros** of **P(x).** We can conclude that:

\[ c = -4\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 4)\ is\ a\ factor\ of\ P(x) \]

\[ c = 3\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 3)\ is\ a\ factor\ of\ P(x) \]

\[ c = 0\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 0)\ is\ a\ factor\ of\ P(x) \]

\[ c = -2\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 2)\ is\ a\ factor\ of\ P(x) \]

We can write that polynomial **P(x)** is equal to the product of its **factors** according to the **factor theorem.** The expression for **P(x)** is given as:

\[ P(x) = ( x + 4 )( x\ -\ 3 )( x\ -\ 0 )( x + 2 ) \]

\[ P(x) = x( x + 2 )( x\ -\ 3 )( x + 4 ) \]

Simplifying the equation will give us the **polynomial P(x).**

\[ P(x) = (x^2 + 2x )( x^2 + x\ -\ 12) \]

\[ P(x) = x^4 + 3x^3\ -\ 10x^2\ -\ 24x \]

## Numerical Result

The **polynomial P(x)** with degree **4** and **zeros -4, 3, 0, and -2** is calculated to be:

\[ P(x) = x^4 + 3x^3\ -\ 10x^2\ -\ 24x \]

## Example

Find a **polynomial** with **degree 3** and** zeros -1, 0, and 1.**

Let **P(x)** is the **polynomial function** with a **degree of 3**. It has zeros of **-1, 0, and 1**. So following must be true for the polynomial **P(x).**

\[ c = -1\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x + 1)\ is\ a\ factor\ of\ P(x) \]

\[ c = 1\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 1)\ is\ a\ factor\ of\ P(x) \]

\[ c = 0\ is\ a\ zero\ of\ P(x) \]

\[ \Rightarrow (x\ -\ 0)\ is\ a\ factor\ of\ P(x) \]

We can write the **P(x)** equal to its **factors** as:

\[ P(x) = x( x + 1 )( x\ -\ 1 ) \]

\[ P(x) = x( x^2\ -\ x + x\ -\ 1 ) \]

\[ P(x) = x( x^2\ -\ 1 ) \]

\[ P(x) = x^3\ -\ x \]

The **polynomial P(x)** has a **degree** of **3**.