Question: Perform the indicated operations and simplify.
(a) $\left(12 - \frac{2}{3}x + \frac{1}{2}x^2\right) - \left(x^3 + 3x^2 - \frac{1}{6}x\right)$
(b) $\left[z + (y + 1)\right]\left[z - (y + 1)\right]$
(c) $\left(6x^m - 5\right)\left(2x^{2m} - 3\right)$
1. **Problem (a):** Simplify $\left(12 - \frac{2}{3}x + \frac{1}{2}x^2\right) - \left(x^3 + 3x^2 - \frac{1}{6}x\right)$.
2. **Step 1:** Distribute the minus sign to the second parentheses:
$$12 - \frac{2}{3}x + \frac{1}{2}x^2 - x^3 - 3x^2 + \frac{1}{6}x$$
3. **Step 2:** Group like terms:
$$-x^3 + \left(\frac{1}{2}x^2 - 3x^2\right) + \left(-\frac{2}{3}x + \frac{1}{6}x\right) + 12$$
4. **Step 3:** Simplify each group:
- For $x^2$ terms: $$\frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = -\frac{5}{2}$$
- For $x$ terms: $$-\frac{2}{3} + \frac{1}{6} = -\frac{4}{6} + \frac{1}{6} = -\frac{3}{6} = -\frac{1}{2}$$
5. **Step 4:** Write the simplified expression:
$$-x^3 - \frac{5}{2}x^2 - \frac{1}{2}x + 12$$
---
6. **Problem (b):** Simplify $\left[z + (y + 1)\right]\left[z - (y + 1)\right]$.
7. **Step 1:** Recognize this as a difference of squares:
$$ (a + b)(a - b) = a^2 - b^2 $$
where $a = z$ and $b = y + 1$.
8. **Step 2:** Apply the formula:
$$z^2 - (y + 1)^2$$
9. **Step 3:** Expand the square:
$$(y + 1)^2 = y^2 + 2y + 1$$
10. **Step 4:** Final simplified expression:
$$z^2 - y^2 - 2y - 1$$
---
11. **Problem (c):** Simplify $\left(6x^m - 5\right)\left(2x^{2m} - 3\right)$.
12. **Step 1:** Use distributive property (FOIL):
$$6x^m \cdot 2x^{2m} + 6x^m \cdot (-3) - 5 \cdot 2x^{2m} - 5 \cdot (-3)$$
13. **Step 2:** Multiply terms:
- $$6 \times 2 = 12$$ and $$x^m \times x^{2m} = x^{m + 2m} = x^{3m}$$
- $$6x^m \times (-3) = -18x^m$$
- $$-5 \times 2x^{2m} = -10x^{2m}$$
- $$-5 \times (-3) = 15$$
14. **Step 3:** Write the expanded expression:
$$12x^{3m} - 18x^m - 10x^{2m} + 15$$
15. **Step 4:** Rearrange terms in descending powers:
$$12x^{3m} - 10x^{2m} - 18x^m + 15$$