To answer this question, we need to understand the concept of a "zero" of a polynomial. If $x = 1$ is a zero of the polynomial $P(x) = 3x^3 - 3x^2 + 2x - 2$, it means that when we substitute $x = 1$ into the polynomial, the result must be zero.

Let's check if this is the case:

$P(1) = 3(1)^3 - 3(1)^2 + 2(1) - 2 = 3 - 3 + 2 - 2 = 0$

Since $P(1) = 0$, $x = 1$ is indeed a zero of the polynomial.

The question is asking us to choose the statement that is true about the polynomial based on this fact.

Analyzing the options:

• A. $P(x)$ is divisible by $x - 1$:
  Since $x = 1$ is a zero, by the Factor Theorem, we know that $P(x)$ is divisible by $x - 1$.

• B. Equation $P(x) - 1 = 0$ is satisfied for every value of $x$:
  This is not correct. The equation $P(x) = 1$ would only be true for specific values of $x$, not for all values of $x$.

• C. $P(x)$ is divisible by $x + 1$:
  This is not necessarily true. Since $x = 1$ is a zero, $x + 1$ would not be related to this case.

• D. Equation $P(x) = 0$ has three real roots, one of which is the number 1:
  This is plausible but requires further analysis, such as factoring the polynomial. However, the question doesn't ask for that level of detail.

• E. $P(x)$ is a power of $x - 1$:
  This is not necessarily true. While $x = 1$ is a root, it doesn't mean $P(x)$ is a power of $x - 1$.

Conclusion:

The correct answer is A. $P(x)$ is divisible by $x - 1$.

Would you like further details or explanations on any of these points?

Here are 5 related questions:

1. What does the Factor Theorem state, and how does it apply here?
2. How can we factor a polynomial when we know one of its zeros?
3. What does it mean for a polynomial to have real roots?
4. How can we use synthetic division to check for divisibility by $x - 1$?
5. What is the relationship between the degree of a polynomial and the number of real roots?

Tip: Always verify by substitution if a given number is truly a zero of the polynomial before applying the Factor Theorem!