To solve the problem, we need to use synthetic division to determine if $x-1$ is a factor of the polynomial $f(x) = x^3 - 5x^2 + 12x - 8$. Let's solve it step by step:

1. Set up synthetic division:

   • The divisor is $x - 1$, so we use $1$ as the value for synthetic division.
   • The coefficients of $f(x)$ are $[1, -5, 12, -8]$.

2. Perform synthetic division:

• Bring down the first coefficient ($1$).
• Multiply $1$ by $1$, and add to the next coefficient ($-5$), resulting in $-4$.
• Multiply $-4$ by $1$, and add to $12$, resulting in $8$.
• Multiply $8$ by $1$, and add to $-8$, resulting in $0$.

3. Interpret the remainder:

   • The remainder is $0$, meaning $x-1$ is a factor of $f(x)$.

4. Answer:

   • The correct choice is: "Yes, because the remainder = 0."

Would you like me to explain synthetic division further or show related problems?

Related Questions:

1. What happens if the remainder is not zero in synthetic division?
2. How do you write $f(x)$ as a product of $x-1$ and the quotient?
3. How can you verify that $x-1$ is a factor using substitution?
4. Can you explain synthetic division for divisors like $x + a$?
5. How does synthetic division compare to long division?

Tip: Always check the remainder when testing if a linear factor divides a polynomial—it tells you whether it’s truly a factor!