polynomial with given zeros

1. Understand the Factor Theorem: The Factor Theorem states that a polynomial $P(x)$ has a zero at $x = c$ if and only if $(x - c)$ is a factor of the polynomial. This means that if we know the zeros, we can construct the polynomial by multiplying the corresponding factors.

2. Identify the factors from the given zeros: The problem states that the polynomial has zeros at $x = 1$ and $x = 6$. According to the Factor Theorem:
- For the zero at $x = 1$, the corresponding factor is $(x - 1)$.
- For the zero at $x = 6$, the corresponding factor is $(x - 6)$.

3. Construct the polynomial: To create a polynomial with these specific zeros, we multiply the factors together:
$$P(x) = (x - 1)(x - 6)$$

4. Verify the options:
- $(x + 1)^6$: This has a zero at $x = -1$.
- $(x - 1)^6$: This has a zero at $x = 1$ (with multiplicity 6), but not at $x = 6$.
- $(x - 1)(x - 6)$: This has zeros at $x = 1$ and $x = 6$, which matches the requirement.
- $(x + 1)(x + 6)$: This has zeros at $x = -1$ and $x = -6$.