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The question asks to find all the roots of the function \( f(x) = x^3 + 3x^2 - x - 3 \) given that \( f(1) = 0 \), and we are to use the Remainder Theorem.

### Steps:
1. **Apply the Remainder Theorem**: According to the Remainder Theorem, if \( f(1) = 0 \), then \( x - 1 \) is a factor of \( f(x) \).

2. **Factorize the function**:
   We will divide \( f(x) = x^3 + 3x^2 - x - 3 \) by \( x - 1 \) using synthetic or polynomial division to find the quotient.

3. **Synthetic Division**:
   - Set up synthetic division with the divisor \( x - 1 \).
   - The coefficients of \( f(x) \) are \( [1, 3, -1, -3] \).

Performing the synthetic division:

\[
\begin{array}{r|rrrr}
1 & 1 & 3 & -1 & -3 \\
  &   & 1 & 4  & 3 \\
\hline
  & 1 & 4 & 3  & 0 \\
\end{array}
\]

The quotient is \( x^2 + 4x + 3 \), and there is no remainder.

4. **Factor the quotient**: Now, factor \( x^2 + 4x + 3 \).

\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]

5. **Find all the roots**: The fully factored form of \( f(x) \) is:

\[
f(x) = (x - 1)(x + 1)(x + 3)
\]

Setting each factor equal to zero gives the roots:
- \( x = 1 \)
- \( x = -1 \)
- \( x = -3 \)

### Conclusion:
The roots of the function are \( x = 1 \), \( x = -1 \), and \( x = -3 \).

Thus, the correct answer is:
\[
\boxed{x = -3, x = -1, x = 1}
\]

Let me know if you'd like further clarification!

### Follow-up Questions:
1. How does the Remainder Theorem help in finding the roots of a polynomial?
2. What is synthetic division, and how is it applied to factor polynomials?
3. Can all cubic polynomials be factorized using simple methods like synthetic division?
4. What is the significance of finding all roots of a polynomial?
5. How would you check if the factored form of the polynomial is correct?

### Tip:
When using the Remainder Theorem, always remember that if \( f(a) = 0 \), then \( x - a \) is a factor of the polynomial.

If f(1) = 0, what are all the roots of the function f(x) = x^3 + 3x^2 - x - 3? Use the Remainder Theorem.

Explore the process of finding all the roots of the polynomial f(x) = x^3 + 3x^2 - x - 3 using the Remainder Theorem and polynomial division. This algebra problem involves factoring and determining the roots for students in Grades 9-12.

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