Let's work through these polynomial equations step by step. I'll solve Part A first and explain each step clearly.

A. Solve each equation

1. i. $x - 3 = 0$:
   Add $3$ to both sides:
   $x = 3$

2. ii. $x + 4 = 0$:
   Subtract $4$ from both sides:
   $x = -4$

3. iii. $3z + 2 = 0$:
   Subtract $2$ and divide by $3$:
   $z = -\frac{2}{3}$

4. iv. $-2z + 4 = 0$:
   Subtract $4$, then divide by $-2$:
   $z = 2$

5. v. $7x = 0$:
   Divide by $7$:
   $x = 0$

6. vi. $\frac{1}{2}z = 0$:
   Multiply both sides by $2$:
   $z = 0$

B. Use results from Part A

Here we solve each equation in Part B by factoring and using the Zero Product Property. This property states that if $ab = 0$, then $a = 0$ or $b = 0$.

1. i. $2(x - 3) = 0$:
   Since $2 \neq 0$, the solution comes from $x - 3 = 0$:
   $x = 3$

2. ii. $-12(x + 4) = 0$:
   Since $-12 \neq 0$, the solution comes from $x + 4 = 0$:
   $x = -4$

3. iii. $7x(x + 4) = 0$:
   Set each factor equal to $0$:
   $7x = 0 \quad \text{or} \quad x + 4 = 0$
   Solve:
   $x = 0 \quad \text{or} \quad x = -4$

4. iv. $\frac{1}{2}z(3z + 2) = 0$:
   Set each factor equal to $0$:
   $\frac{1}{2}z = 0 \quad \text{or} \quad 3z + 2 = 0$
   Solve:
   $z = 0 \quad \text{or} \quad z = -\frac{2}{3}$

5. v. $(x - 3)(x + 4) = 0$:
   Set each factor equal to $0$:
   $x - 3 = 0 \quad \text{or} \quad x + 4 = 0$
   Solve:
   $x = 3 \quad \text{or} \quad x = -4$

6. vi. $(3z + 2)(-2z + 4) = 0$:
   Set each factor equal to $0$:
   $3z + 2 = 0 \quad \text{or} \quad -2z + 4 = 0$
   Solve:
   $z = -\frac{2}{3} \quad \text{or} \quad z = 2$

Final Answers:

• A (part i to vi):
  $x = 3, \, x = -4, \, z = -\frac{2}{3}, \, z = 2, \, x = 0, \, z = 0$

• B (part i to vi):
  $x = 3, \, x = -4, \, x = 0 \, \text{or} \, -4, \, z = 0 \, \text{or} \, -\frac{2}{3}, \, x = 3 \, \text{or} \, -4, \, z = -\frac{2}{3} \, \text{or} \, 2$

Do you want me to clarify any step or show additional explanations?

Related Questions:

1. How do you solve polynomial equations using factoring?
2. What is the Zero Product Property and why is it important?
3. How can you solve higher-degree polynomials using factoring?
4. What are some tips for solving equations with fractions or coefficients?
5. How can you verify solutions for polynomial equations?

Tip: Always set each factor of a factored equation equal to $0$ to solve polynomial equations efficiently.