Use this worksheet to practice fundamental polynomial operations including addition, subtraction, multiplication, and division.

Problem: Solve the polynomial expressions as instructed in the worksheet.

#### Part 1: Add the given expressions

1. Expression: $(6 + 2n^3) + (5n^2 + 2)$
- Combine like terms:
\[
6 + 2n^3 + 5n^2 + 2 = 2n^3 + 5n^2 + (6 + 2) = 2n^3 + 5n^2 + 8
\]
- Answer: $\boxed{2n^3 + 5n^2 + 8}$

2. Expression: $(6m^5 + 1) + (4m^5 + 9m - 1)$
- Combine like terms:
\[
6m^5 + 1 + 4m^5 + 9m - 1 = (6m^5 + 4m^5) + 9m + (1 - 1) = 10m^5 + 9m
\]
- Answer: $\boxed{10m^5 + 9m}$

3. Expression: $(3y^2 - y) + 5y^3 + (-4y^3 + y^2 - 8)$
- Combine like terms:
\[
3y^2 - y + 5y^3 - 4y^3 + y^2 - 8 = (5y^3 - 4y^3) + (3y^2 + y^2) - y - 8 = y^3 + 4y^2 - y - 8
\]
- Answer: $\boxed{y^3 + 4y^2 - y - 8}$

4. Expression: $(4v^4 - v^2) + (5v^2 + v^4)$
- Combine like terms:
\[
4v^4 - v^2 + 5v^2 + v^4 = (4v^4 + v^4) + (-v^2 + 5v^2) = 5v^4 + 4v^2
\]
- Answer: $\boxed{5v^4 + 4v^2}$

#### Part 2: Subtract the given expressions

5. Expression: $(-2x^3 + x) - (7x - 3 + 7x^3)$
- Distribute the negative sign and combine like terms:
\[
-2x^3 + x - 7x + 3 - 7x^3 = (-2x^3 - 7x^3) + (x - 7x) + 3 = -9x^3 - 6x + 3
\]
- Answer: $\boxed{-9x^3 - 6x + 3}$

6. Expression: $(3a^2 + 2a - 2) - (a^2 - 3a + 7)$
- Distribute the negative sign and combine like terms:
\[
3a^2 + 2a - 2 - a^2 + 3a - 7 = (3a^2 - a^2) + (2a + 3a) + (-2 - 7) = 2a^2 + 5a - 9
\]
- Answer: $\boxed{2a^2 + 5a - 9}$

#### Part 3: Multiply the given expressions

7. Expression: $(4v - 2)(7v^2 + 6v - 8)$
- Use the distributive property (FOIL method for polynomials):
\[
(4v - 2)(7v^2 + 6v - 8) = 4v(7v^2 + 6v - 8) - 2(7v^2 + 6v - 8)
\]
\[
= 4v \cdot 7v^2 + 4v \cdot 6v + 4v \cdot (-8) - 2 \cdot 7v^2 - 2 \cdot 6v - 2 \cdot (-8)
\]
\[
= 28v^3 + 24v^2 - 32v - 14v^2 - 12v + 16
\]
- Combine like terms:
\[
28v^3 + (24v^2 - 14v^2) + (-32v - 12v) + 16 = 28v^3 + 10v^2 - 44v + 16
\]
- Answer: $\boxed{28v^3 + 10v^2 - 44v + 16}$

8. Expression: $(4x^2 + 3x + 1)(7x^2 - 4x - 2)$
- Use the distributive property:
\[
(4x^2 + 3x + 1)(7x^2 - 4x - 2) = 4x^2(7x^2 - 4x - 2) + 3x(7x^2 - 4x - 2) + 1(7x^2 - 4x - 2)
\]
\[
= 4x^2 \cdot 7x^2 + 4x^2 \cdot (-4x) + 4x^2 \cdot (-2) + 3x \cdot 7x^2 + 3x \cdot (-4x) + 3x \cdot (-2) + 1 \cdot 7x^2 + 1 \cdot (-4x) + 1 \cdot (-2)
\]
\[
= 28x^4 - 16x^3 - 8x^2 + 21x^3 - 12x^2 - 6x + 7x^2 - 4x - 2
\]
- Combine like terms:
\[
28x^4 + (-16x^3 + 21x^3) + (-8x^2 - 12x^2 + 7x^2) + (-6x - 4x) - 2 = 28x^4 + 5x^3 - 13x^2 - 10x - 2
\]
- Answer: $\boxed{28x^4 + 5x^3 - 13x^2 - 10x - 2}$

#### Part 4: Divide the given expressions

9. Expression: $(n^2 - n - 80) \div (n - 9)$
- Perform polynomial long division:
\[
\begin{array}{r|rr}
n - 9 & n^2 & -n & -80 \\
& n^2 & -9n & \\
\hline
& & 8n & -80 \\
& & 8n & -72 \\
\hline
& & & -8 \\
\end{array}
\]
- Quotient: $n + 8$
- Remainder: $-8$
- Therefore, the result is:
\[
n + 8 - \frac{8}{n - 9}
\]
- Answer: $\boxed{n + 8 - \frac{8}{n - 9}}$

10. Expression: $(2x^{11} - 5x^7 - 10x^6) \div 2x^3$
- Divide each term by $2x^3$:
\[
\frac{2x^{11}}{2x^3} - \frac{5x^7}{2x^3} - \frac{10x^6}{2x^3} = x^{11-3} - \frac{5}{2}x^{7-3} - 5x^{6-3}
\]
\[
= x^8 - \frac{5}{2}x^4 - 5x^3
\]
- Answer: $\boxed{x^8 - \frac{5}{2}x^4 - 5x^3}$

Final Answers:

1. $\boxed{2n^3 + 5n^2 + 8}$
2. $\boxed{10m^5 + 9m}$
3. $\boxed{y^3 + 4y^2 - y - 8}$
4. $\boxed{5v^4 + 4v^2}$
5. $\boxed{-9x^3 - 6x + 3}$
6. $\boxed{2a^2 + 5a - 9}$
7. $\boxed{28v^3 + 10v^2 - 44v + 16}$
8. $\boxed{28x^4 + 5x^3 - 13x^2 - 10x - 2}$
9. $\boxed{n + 8 - \frac{8}{n - 9}}$
10. $\boxed{x^8 - \frac{5}{2}x^4 - 5x^3}$