`ax^3 + bx^2+cx +d =0`
In the equation above, a, b, c, and d are constants. If the equation has roots −1, −3, and 5, which of the following is a factor of `ax^3 + bx^2 + cx + d` ?

Options

A. x-1
B. x+1
C. x − 3
D. x+5

Explanation

To determine which of the following is a factor of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) given that the roots are \( -1, -3, \) and \( 5 \), we can follow these steps:

### Step 1: Understand the relationship between roots and factors
If a polynomial has roots \( r_1, r_2, \) and \( r_3 \), then the polynomial can be expressed in factored form as:
\[ a(x - r_1)(x - r_2)(x - r_3) \]
where \( a \) is a non-zero constant.

### Step 2: Identify the roots
The roots given in the problem are:
- \( r_1 = -1 \)
- \( r_2 = -3 \)
- \( r_3 = 5 \)

### Step 3: Write the factors corresponding to the roots
From the roots, we can write the factors:
- For \( r_1 = -1 \), the factor is \( (x + 1) \)
- For \( r_2 = -3 \), the factor is \( (x + 3) \)
- For \( r_3 = 5 \), the factor is \( (x - 5) \)

### Step 4: List the factors
Thus, the polynomial can be expressed as:
\[ a(x + 1)(x + 3)(x - 5) \]
This means that the factors of the polynomial \( ax^3 + bx^2 + cx + d \) include \( (x + 1) \), \( (x + 3) \), and \( (x - 5) \).

### Step 5: Identify the correct option
Now, we need to check which of the provided options is a factor:
- Option a: \( x - 1 \) (not a factor)
- Option b: \( x + 1 \) (is a factor)
- Option c: \( x - 3 \) (not a factor)
- Option d: \( x + 5 \) (not a factor)

The correct answer is **Option b: \( x + 1 \)**.