We are tasked with finding the remainder when $f(x) = 2x^3 - 3x^2 - 6x - 7$ is divided by $x - 3$ using the Remainder Theorem and then using the Factor Theorem to check if $x - 3$ is a factor.

Step 1: Use the Remainder Theorem

The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - a$, the remainder is $f(a)$. Here, we divide by $x - 3$, so we calculate $f(3)$.

Given $f(x) = 2x^3 - 3x^2 - 6x - 7$, we substitute $x = 3$ into the function:

$f(3) = 2(3)^3 - 3(3)^2 - 6(3) - 7$

Step 2: Simplify the expression

$f(3) = 2(27) - 3(9) - 6(3) - 7$ $f(3) = 54 - 27 - 18 - 7$ $f(3) = 54 - 52 = 2$

Step 3: Conclusion

• The remainder when $f(x)$ is divided by $x - 3$ is 2.
• According to the Factor Theorem, since $f(3) \neq 0$, $x - 3$ is not a factor of $f(x)$.

Would you like further details or have any questions about the solution?

Here are five related questions to expand on this topic:

1. How do you apply the Remainder Theorem to other polynomials?
2. What is the Factor Theorem and how is it related to the Remainder Theorem?
3. Can you explain how synthetic division could be used to find the remainder?
4. How do you determine if a higher-degree polynomial has multiple factors like $x - 3$?
5. What are the practical applications of using the Remainder Theorem?

Tip: When using the Remainder Theorem, make sure to evaluate the polynomial at the appropriate root for accurate results.