If 4 and 0 are both solutions to the equation q(x) = 0, where q(x) is a polynomial, then it can be concluded that a factor of q(x) is

Options

A. `x^(2)`
B. `(x-4)^(2)`
C. `x^(2)+4x`
D. `x^(2)-4x`

Explanation

To find a factor of the polynomial \( q(x) \) given that 4 and 0 are solutions to the equation \( q(x) = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Solutions**:
Since 4 and 0 are solutions to the equation \( q(x) = 0 \), it means that when \( x = 4 \) and \( x = 0 \), the polynomial \( q(x) \) evaluates to zero.

2. **Writing Factors from Solutions**:
From the solution \( x = 4 \), we can derive a factor of the polynomial:
\[ x - 4 = 0 \quad \text{(this implies that } x = 4\text{ is a root)} \]
From the solution \( x = 0 \), we can derive another factor:
\[ x - 0 = x \quad \text{(this implies that } x = 0\text{ is a root)} \]

3. **Forming the Polynomial**:
Since both \( x - 4 \) and \( x \) are factors of \( q(x) \), we can express \( q(x) \) as:
\[ q(x) = k \cdot (x)(x - 4) \]
where \( k \) is a constant (which can be any non-zero number).

4. **Multiplying the Factors**:
To find a specific form of \( q(x) \), we can multiply the factors:
\[ q(x) = k \cdot (x)(x - 4) = k \cdot (x^2 - 4x) \]
This shows that \( x^2 - 4x \) is a polynomial that has 4 and 0 as roots.

5. **Conclusion**:
Therefore, we can conclude that a factor of \( q(x) \) is:
\[ x(x - 4) \]
or simply \( x^2 - 4x \) if we consider \( k = 1 \).

### Final Answer:
A factor of \( q(x) \) is \( x(x - 4) \) or \( x^2 - 4x \).

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