Students practice substituting values into linear, polynomial, and piecewise function equations.

Problem Analysis:

The task involves evaluating the function \( q(x) = x^3 - x^2 \) for various inputs. The function is a polynomial, and we need to substitute the given values of \( x \) into the function and simplify the result.

The function is:
\[
q(x) = x^3 - x^2
\]

We will evaluate \( q(x) \) for the following inputs:
1. \( q(-3) \)
2. \( q(0) \)
3. \( q(5) \)
4. \( q(-2) \)
5. \( q(1) \)
6. \( q(3) \)
7. \( q(0) \)
8. \( q(1) \)

Step-by-Step Solution:

#### 1. Evaluate \( q(-3) \):
\[
q(-3) = (-3)^3 - (-3)^2
\]
Calculate each term:
\[
(-3)^3 = -27 \quad \text{and} \quad (-3)^2 = 9
\]
Substitute back:
\[
q(-3) = -27 - 9 = -36
\]

#### 2. Evaluate \( q(0) \):
\[
q(0) = (0)^3 - (0)^2
\]
Calculate each term:
\[
(0)^3 = 0 \quad \text{and} \quad (0)^2 = 0
\]
Substitute back:
\[
q(0) = 0 - 0 = 0
\]

#### 3. Evaluate \( q(5) \):
\[
q(5) = (5)^3 - (5)^2
\]
Calculate each term:
\[
(5)^3 = 125 \quad \text{and} \quad (5)^2 = 25
\]
Substitute back:
\[
q(5) = 125 - 25 = 100
\]

#### 4. Evaluate \( q(-2) \):
\[
q(-2) = (-2)^3 - (-2)^2
\]
Calculate each term:
\[
(-2)^3 = -8 \quad \text{and} \quad (-2)^2 = 4
\]
Substitute back:
\[
q(-2) = -8 - 4 = -12
\]

#### 5. Evaluate \( q(1) \):
\[
q(1) = (1)^3 - (1)^2
\]
Calculate each term:
\[
(1)^3 = 1 \quad \text{and} \quad (1)^2 = 1
\]
Substitute back:
\[
q(1) = 1 - 1 = 0
\]

#### 6. Evaluate \( q(3) \):
\[
q(3) = (3)^3 - (3)^2
\]
Calculate each term:
\[
(3)^3 = 27 \quad \text{and} \quad (3)^2 = 9
\]
Substitute back:
\[
q(3) = 27 - 9 = 18
\]

#### 7. Evaluate \( q(0) \):
This is the same as part 2:
\[
q(0) = 0
\]

#### 8. Evaluate \( q(1) \):
This is the same as part 5:
\[
q(1) = 0
\]

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & q(-3) = -36 \\
2. & q(0) = 0 \\
3. & q(5) = 100 \\
4. & q(-2) = -12 \\
5. & q(1) = 0 \\
6. & q(3) = 18 \\
7. & q(0) = 0 \\
8. & q(1) = 0 \\
\end{array}
}
\]