In the equation `ax^(4)+bx ^(3)+cx ^(2) -dx=0,a,b,c and d` are constants. If the equations crosses the x-axis at `0,-2,3,and 5` which of the following is a factor of `ax ^(4)+bx ^(3)+cx ^(2)-dx` ?

Options

A. `x -2`
B. `x +3`
C. `x -5`
D. `x +5`

Answer

To solve the problem, we need to determine which of the given options is a factor of the polynomial \( ax^4 + bx^3 + cx^2 - dx = 0 \) based on the roots provided. The roots are \( 0, -2, 3, \) and \( 5 \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
The polynomial crosses the x-axis at the points \( x = 0, -2, 3, \) and \( 5 \). This means that these values are the roots of the polynomial.

2. **Identifying Factors from Roots**:
According to the factor theorem, if \( x = r \) is a root of a polynomial \( P(x) \), then \( (x - r) \) is a factor of \( P(x) \).
- For \( x = 0 \): The factor is \( (x - 0) = x \).
- For \( x = -2 \): The factor is \( (x + 2) \).
- For \( x = 3 \): The factor is \( (x - 3) \).
- For \( x = 5 \): The factor is \( (x - 5) \).

3. **Listing All Factors**:
From the roots, we can list the factors:
- \( x \) (from \( x = 0 \))
- \( x + 2 \) (from \( x = -2 \))
- \( x - 3 \) (from \( x = 3 \))
- \( x - 5 \) (from \( x = 5 \))

4. **Analyzing the Options**:
The options given are:
- A) \( x - 2 \)
- B) \( x + 3 \)
- C) \( x - 5 \)
- D) \( x + 5 \)

5. **Determining the Correct Factor**:
From our list of factors, we see that \( x - 5 \) is indeed one of the factors derived from the roots. Therefore, the correct answer is:
- **C) \( x - 5 \)**

### Final Answer:
The factor of the polynomial \( ax^4 + bx^3 + cx^2 - dx \) is \( x - 5 \).