Dividing Polynomials worksheet featuring eight algebraic division problems for single variable expressions.

Problem: Dividing Polynomials

We are tasked with dividing the given polynomials. Let's solve each problem step by step.

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#### 1) \( (6x^2 + 3x^8 + 9x^7) \div 3x \)

Step 1: Write the division as a fraction:
\[
\frac{6x^2 + 3x^8 + 9x^7}{3x}
\]

Step 2: Divide each term in the numerator by \(3x\):
\[
\frac{6x^2}{3x} + \frac{3x^8}{3x} + \frac{9x^7}{3x}
\]

Step 3: Simplify each term:
\[
\frac{6x^2}{3x} = 2x^{2-1} = 2x
\]
\[
\frac{3x^8}{3x} = x^{8-1} = x^7
\]
\[
\frac{9x^7}{3x} = 3x^{7-1} = 3x^6
\]

Step 4: Combine the results:
\[
2x + x^7 + 3x^6
\]

Final Answer:
\[
\boxed{2x + x^7 + 3x^6}
\]

---

#### 2) \( (2k^5 - 8k^7 - 6k^2) \div 2k^2 \)

Step 1: Write the division as a fraction:
\[
\frac{2k^5 - 8k^7 - 6k^2}{2k^2}
\]

Step 2: Divide each term in the numerator by \(2k^2\):
\[
\frac{2k^5}{2k^2} + \frac{-8k^7}{2k^2} + \frac{-6k^2}{2k^2}
\]

Step 3: Simplify each term:
\[
\frac{2k^5}{2k^2} = k^{5-2} = k^3
\]
\[
\frac{-8k^7}{2k^2} = -4k^{7-2} = -4k^5
\]
\[
\frac{-6k^2}{2k^2} = -3
\]

Step 4: Combine the results:
\[
k^3 - 4k^5 - 3
\]

Final Answer:
\[
\boxed{k^3 - 4k^5 - 3}
\]

---

#### 3) \( (5m^4 + 10m^2 - 15) \div 5 \)

Step 1: Write the division as a fraction:
\[
\frac{5m^4 + 10m^2 - 15}{5}
\]

Step 2: Divide each term in the numerator by \(5\):
\[
\frac{5m^4}{5} + \frac{10m^2}{5} + \frac{-15}{5}
\]

Step 3: Simplify each term:
\[
\frac{5m^4}{5} = m^4
\]
\[
\frac{10m^2}{5} = 2m^2
\]
\[
\frac{-15}{5} = -3
\]

Step 4: Combine the results:
\[
m^4 + 2m^2 - 3
\]

Final Answer:
\[
\boxed{m^4 + 2m^2 - 3}
\]

---

#### 4) \( (-8n^6 - n^3) \div (-n^3) \)

Step 1: Write the division as a fraction:
\[
\frac{-8n^6 - n^3}{-n^3}
\]

Step 2: Divide each term in the numerator by \(-n^3\):
\[
\frac{-8n^6}{-n^3} + \frac{-n^3}{-n^3}
\]

Step 3: Simplify each term:
\[
\frac{-8n^6}{-n^3} = 8n^{6-3} = 8n^3
\]
\[
\frac{-n^3}{-n^3} = 1
\]

Step 4: Combine the results:
\[
8n^3 + 1
\]

Final Answer:
\[
\boxed{8n^3 + 1}
\]

---

#### 5) \( (-4w^7 + 6w^6 + w^3 - 5w^2) \div w^3 \)

Step 1: Write the division as a fraction:
\[
\frac{-4w^7 + 6w^6 + w^3 - 5w^2}{w^3}
\]

Step 2: Divide each term in the numerator by \(w^3\):
\[
\frac{-4w^7}{w^3} + \frac{6w^6}{w^3} + \frac{w^3}{w^3} + \frac{-5w^2}{w^3}
\]

Step 3: Simplify each term:
\[
\frac{-4w^7}{w^3} = -4w^{7-3} = -4w^4
\]
\[
\frac{6w^6}{w^3} = 6w^{6-3} = 6w^3
\]
\[
\frac{w^3}{w^3} = 1
\]
\[
\frac{-5w^2}{w^3} = -5w^{2-3} = -5w^{-1} = -\frac{5}{w}
\]

Step 4: Combine the results:
\[
-4w^4 + 6w^3 + 1 - \frac{5}{w}
\]

Final Answer:
\[
\boxed{-4w^4 + 6w^3 + 1 - \frac{5}{w}}
\]

---

#### 6) \( (-7x^5 + 7x^3) \div 7x^4 \)

Step 1: Write the division as a fraction:
\[
\frac{-7x^5 + 7x^3}{7x^4}
\]

Step 2: Divide each term in the numerator by \(7x^4\):
\[
\frac{-7x^5}{7x^4} + \frac{7x^3}{7x^4}
\]

Step 3: Simplify each term:
\[
\frac{-7x^5}{7x^4} = -x^{5-4} = -x
\]
\[
\frac{7x^3}{7x^4} = x^{3-4} = x^{-1} = \frac{1}{x}
\]

Step 4: Combine the results:
\[
-x + \frac{1}{x}
\]

Final Answer:
\[
\boxed{-x + \frac{1}{x}}
\]

---

#### 7) \( (8y^3 + 16y^2) \div (-8y) \)

Step 1: Write the division as a fraction:
\[
\frac{8y^3 + 16y^2}{-8y}
\]

Step 2: Divide each term in the numerator by \(-8y\):
\[
\frac{8y^3}{-8y} + \frac{16y^2}{-8y}
\]

Step 3: Simplify each term:
\[
\frac{8y^3}{-8y} = -y^{3-1} = -y^2
\]
\[
\frac{16y^2}{-8y} = -2y^{2-1} = -2y
\]

Step 4: Combine the results:
\[
-y^2 - 2y
\]

Final Answer:
\[
\boxed{-y^2 - 2y}
\]

---

#### 8) \( (6v^8 - 5v^4 + v^3) \div v \)

Step 1: Write the division as a fraction:
\[
\frac{6v^8 - 5v^4 + v^3}{v}
\]

Step 2: Divide each term in the numerator by \(v\):
\[
\frac{6v^8}{v} + \frac{-5v^4}{v} + \frac{v^3}{v}
\]

Step 3: Simplify each term:
\[
\frac{6v^8}{v} = 6v^{8-1} = 6v^7
\]
\[
\frac{-5v^4}{v} = -5v^{4-1} = -5v^3
\]
\[
\frac{v^3}{v} = v^{3-1} = v^2
\]

Step 4: Combine the results:
\[
6v^7 - 5v^3 + v^2
\]

Final Answer:
\[
\boxed{6v^7 - 5v^3 + v^2}
\]

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Final Answers:

1. \(\boxed{2x + x^7 + 3x^6}\)
2. \(\boxed{k^3 - 4k^5 - 3}\)
3. \(\boxed{m^4 + 2m^2 - 3}\)
4. \(\boxed{8n^3 + 1}\)
5. \(\boxed{-4w^4 + 6w^3 + 1 - \frac{5}{w}}\)
6. \(\boxed{-x + \frac{1}{x}}\)
7. \(\boxed{-y^2 - 2y}\)
8. \(\boxed{6v^7 - 5v^3 + v^2}\)