Worksheet for practicing division of polynomials, including monomial and polynomial division exercises.

Let's solve each problem step by step.

---

Problem 1:

Divide the monomials and simplify.
\[
\frac{-8m^8c^4}{4m^4}
\]

Solution:
1. Divide the coefficients: \(\frac{-8}{4} = -2\).
2. Subtract the exponents of like bases:
- For \(m\): \(8 - 4 = 4\).
- For \(c\): \(4 - 0 = 4\) (since \(c^4\) in the numerator and \(c^0\) in the denominator).

Thus, the simplified expression is:
\[
-2m^4c^4
\]

Answer:
\[
\boxed{-2m^4c^4}
\]

---

Problem 2:

Divide the monomials and simplify.
\[
\frac{-8n^3a^3}{12n^2a^2}
\]

Solution:
1. Divide the coefficients: \(\frac{-8}{12} = -\frac{2}{3}\).
2. Subtract the exponents of like bases:
- For \(n\): \(3 - 2 = 1\).
- For \(a\): \(3 - 2 = 1\).

Thus, the simplified expression is:
\[
-\frac{2}{3}na
\]

Answer:
\[
\boxed{-\frac{2}{3}na}
\]

---

Problem 3:

Divide the monomials and simplify.
\[
\frac{9m^4a^4}{12m^2a^2}
\]

Solution:
1. Divide the coefficients: \(\frac{9}{12} = \frac{3}{4}\).
2. Subtract the exponents of like bases:
- For \(m\): \(4 - 2 = 2\).
- For \(a\): \(4 - 2 = 2\).

Thus, the simplified expression is:
\[
\frac{3}{4}m^2a^2
\]

Answer:
\[
\boxed{\frac{3}{4}m^2a^2}
\]

---

Problem 4:

Divide.
\[
(-n^4c^2 - 4n^2c^2 + n^3c^3) \div (11nc^3)
\]

Solution:
Perform polynomial division term by term:
1. Divide \(-n^4c^2\) by \(11nc^3\):
\[
\frac{-n^4c^2}{11nc^3} = -\frac{n^{4-1}c^{2-3}}{11} = -\frac{n^3}{11c}
\]
2. Divide \(-4n^2c^2\) by \(11nc^3\):
\[
\frac{-4n^2c^2}{11nc^3} = -\frac{4n^{2-1}c^{2-3}}{11} = -\frac{4n}{11c}
\]
3. Divide \(n^3c^3\) by \(11nc^3\):
\[
\frac{n^3c^3}{11nc^3} = \frac{n^{3-1}c^{3-3}}{11} = \frac{n^2}{11}
\]

Combine the results:
\[
-\frac{n^3}{11c} - \frac{4n}{11c} + \frac{n^2}{11}
\]

Answer:
\[
\boxed{-\frac{n^3}{11c} - \frac{4n}{11c} + \frac{n^2}{11}}
\]

---

Problem 5:

Divide.
\[
(-8b^6c^6 + 8b^5c^6 - b^6c^4) \div (15b^3c)
\]

Solution:
Perform polynomial division term by term:
1. Divide \(-8b^6c^6\) by \(15b^3c\):
\[
\frac{-8b^6c^6}{15b^3c} = -\frac{8b^{6-3}c^{6-1}}{15} = -\frac{8b^3c^5}{15}
\]
2. Divide \(8b^5c^6\) by \(15b^3c\):
\[
\frac{8b^5c^6}{15b^3c} = \frac{8b^{5-3}c^{6-1}}{15} = \frac{8b^2c^5}{15}
\]
3. Divide \(-b^6c^4\) by \(15b^3c\):
\[
\frac{-b^6c^4}{15b^3c} = -\frac{b^{6-3}c^{4-1}}{15} = -\frac{b^3c^3}{15}
\]

Combine the results:
\[
-\frac{8b^3c^5}{15} + \frac{8b^2c^5}{15} - \frac{b^3c^3}{15}
\]

Answer:
\[
\boxed{-\frac{8b^3c^5}{15} + \frac{8b^2c^5}{15} - \frac{b^3c^3}{15}}
\]

---

Problem 6:

Divide.
\[
(8n^2m^5 + 7n^5m^5 + 11n^3m^4) \div (5nm^4)
\]

Solution:
Perform polynomial division term by term:
1. Divide \(8n^2m^5\) by \(5nm^4\):
\[
\frac{8n^2m^5}{5nm^4} = \frac{8n^{2-1}m^{5-4}}{5} = \frac{8nm}{5}
\]
2. Divide \(7n^5m^5\) by \(5nm^4\):
\[
\frac{7n^5m^5}{5nm^4} = \frac{7n^{5-1}m^{5-4}}{5} = \frac{7n^4m}{5}
\]
3. Divide \(11n^3m^4\) by \(5nm^4\):
\[
\frac{11n^3m^4}{5nm^4} = \frac{11n^{3-1}m^{4-4}}{5} = \frac{11n^2}{5}
\]

Combine the results:
\[
\frac{8nm}{5} + \frac{7n^4m}{5} + \frac{11n^2}{5}
\]

Answer:
\[
\boxed{\frac{8nm}{5} + \frac{7n^4m}{5} + \frac{11n^2}{5}}
\]

---

Problem 7:

Use long division to divide the polynomials.
\[
(2n^2 - 15n - 26) \div (n - 9)
\]

Solution:
Perform long division:
1. Divide the leading term \(2n^2\) by \(n\):
\[
\frac{2n^2}{n} = 2n
\]
2. Multiply \(2n\) by \(n - 9\):
\[
2n(n - 9) = 2n^2 - 18n
\]
3. Subtract:
\[
(2n^2 - 15n - 26) - (2n^2 - 18n) = 3n - 26
\]
4. Divide the leading term \(3n\) by \(n\):
\[
\frac{3n}{n} = 3
\]
5. Multiply \(3\) by \(n - 9\):
\[
3(n - 9) = 3n - 27
\]
6. Subtract:
\[
(3n - 26) - (3n - 27) = 1
\]

The quotient is \(2n + 3\) and the remainder is \(1\). Thus, the result is:
\[
2n + 3 + \frac{1}{n - 9}
\]

Answer:
\[
\boxed{2n + 3 + \frac{1}{n - 9}}
\]

---

Problem 8:

Use long division to divide the polynomials.
\[
(3x^2 - 14x - 48) \div (x - 7)
\]

Solution:
Perform long division:
1. Divide the leading term \(3x^2\) by \(x\):
\[
\frac{3x^2}{x} = 3x
\]
2. Multiply \(3x\) by \(x - 7\):
\[
3x(x - 7) = 3x^2 - 21x
\]
3. Subtract:
\[
(3x^2 - 14x - 48) - (3x^2 - 21x) = 7x - 48
\]
4. Divide the leading term \(7x\) by \(x\):
\[
\frac{7x}{x} = 7
\]
5. Multiply \(7\) by \(x - 7\):
\[
7(x - 7) = 7x - 49
\]
6. Subtract:
\[
(7x - 48) - (7x - 49) = 1
\]

The quotient is \(3x + 7\) and the remainder is \(1\). Thus, the result is:
\[
3x + 7 + \frac{1}{x - 7}
\]

Answer:
\[
\boxed{3x + 7 + \frac{1}{x - 7}}
\]

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Final Answers:

1. \(\boxed{-2m^4c^4}\)
2. \(\boxed{-\frac{2}{3}na}\)
3. \(\boxed{\frac{3}{4}m^2a^2}\)
4. \(\boxed{-\frac{n^3}{11c} - \frac{4n}{11c} + \frac{n^2}{11}}\)
5. \(\boxed{-\frac{8b^3c^5}{15} + \frac{8b^2c^5}{15} - \frac{b^3c^3}{15}}\)
6. \(\boxed{\frac{8nm}{5} + \frac{7n^4m}{5} + \frac{11n^2}{5}}\)
7. \(\boxed{2n + 3 + \frac{1}{n - 9}}\)
8. \(\boxed{3x + 7 + \frac{1}{x - 7}}\)