To find the zeros of the function f(x) algebraically, we can use the factor theorem and synthetic division.
Given that x - 2 is a factor of f(x), we know that when x = 2, f(x) will equal zero. Using this information, we can perform synthetic division to find the remaining quadratic factor.
Let's start with the synthetic division:
```
2 | 1 -10 7 18
| 2 -16 -18
_______________________
1 -8 -9 0
```
The result of the synthetic division is 1 - 8x - 9x^2 + 0x^3, which corresponds to the remaining factor after dividing by x - 2.
Now, we can set this quadratic factor equal to zero and solve for x:
-9x^2 - 8x + 1 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. In this case, factoring can be used:
(-9x + 1)(x + 1) = 0
From this equation, we can find two potential zeros:
-9x + 1 = 0 --> x = 1/9
x + 1 = 0 --> x = -1
So, the zeros of the function f(x) are x = 2, x = 1/9, and x = -1.
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