Let f(x) be a polynomial function: `f(x)=x^(5)+ . . . .` if f(1)=0 and f(2)=0, then f(x) is divisible by

Options

A. `x-3`
B. `x^(2)-2`
C. `x^(2)+2`
D. `x^(2)-3x+2`

Explanation

To solve the problem, we need to determine what polynomial function \( f(x) \) is divisible by, given that \( f(1) = 0 \) and \( f(2) = 0 \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that \( f(x) \) is a polynomial of degree 5, and it has roots at \( x = 1 \) and \( x = 2 \) because \( f(1) = 0 \) and \( f(2) = 0 \).

**Hint:** A polynomial \( f(x) \) can be expressed in terms of its roots. If \( r \) is a root, then \( (x - r) \) is a factor of the polynomial.

2. **Identifying Factors:**
Since \( f(1) = 0 \), it implies that \( (x - 1) \) is a factor of \( f(x) \). Similarly, since \( f(2) = 0 \), it implies that \( (x - 2) \) is also a factor of \( f(x) \).

**Hint:** If a polynomial has roots at certain values, the factors corresponding to those roots can be directly derived.

3. **Combining the Factors:**
Therefore, we can say that \( f(x) \) is divisible by the product of these factors:
\[ f(x) \text{ is divisible by } (x - 1)(x - 2). \]

**Hint:** To find the polynomial that is divisible by these factors, multiply them together.

4. **Calculating the Product:**
Now, let's calculate the product:
\[ (x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2. \]

**Hint:** Use the distributive property (FOIL method) to multiply the two binomials.

5. **Conclusion:**
Since \( f(x) \) is a polynomial of degree 5, it can be expressed as:
\[ f(x) = (x - 1)(x - 2)g(x), \]
where \( g(x) \) is another polynomial of degree 3. Therefore, \( f(x) \) is divisible by \( x^2 - 3x + 2 \).

**Hint:** The degree of \( g(x) \) is determined by subtracting the degree of the factors from the total degree of the polynomial.

### Final Answer:
Thus, \( f(x) \) is divisible by \( x^2 - 3x + 2 \).