Simplify: $\left(-9 C^4 - 5 C^3 - C^2\right) \div C^3$A) $-9 C + 5 + \frac{1}{c}$B) $-9 C + 1 + \frac{1}{c}$C) $-\frac{1}{9} C + \frac{1}{5} + \frac{1}{c}$D) $-9 C - 5 - \frac{1}{c}$

Simplify: $\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3$

Understanding the Problem

When dealing with polynomial division, it's essential to remember the rules of exponents and the order of operations. In this problem, we're given a polynomial expression $\left(-9 c^4 - 5 c^3 - c^2\right)$ and asked to divide it by $c^3$. To simplify this expression, we need to apply the rules of exponents and perform the division.

Applying the Rules of Exponents

To simplify the expression, we need to apply the rule of exponents that states when dividing like bases, we subtract the exponents. In this case, we have:

$$\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3 = -9 c^{4-3} - 5 c^{3-3} - c^{2-3}$$

Simplifying the exponents, we get:

$$-9 c^1 - 5 c^0 - c^{-1}$$

Simplifying the Expression

Now that we have simplified the exponents, we can simplify the expression further by applying the rules of arithmetic operations. We have:

$$-9 c^1 - 5 c^0 - c^{-1}$$

Since $c^0 = 1$, we can simplify the expression as:

$$-9 c - 5 - \frac{1}{c}$$

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options to see which one matches our result.

A) $-9 c + 5 + \frac{1}{c}$

B) $-9 c + 1 + \frac{1}{c}$

C) $-\frac{1}{9} c + \frac{1}{5} + \frac{1}{c}$

D) $-9 c - 5 - \frac{1}{c}$

Comparing our result with the options, we can see that option D) $-9 c - 5 - \frac{1}{c}$ matches our result.

Conclusion

In this problem, we simplified the expression $\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3$ by applying the rules of exponents and performing the division. We found that the simplified expression is $-9 c - 5 - \frac{1}{c}$. This result matches option D) $-9 c - 5 - \frac{1}{c}$.

Key Takeaways

• When dividing like bases, we subtract the exponents.
• When simplifying the expression, we need to apply the rules of arithmetic operations.
• The simplified expression is $-9 c - 5 - \frac{1}{c}$.

Practice Problems

• Simplify the expression $\left(2 x^3 + 3 x^2 + x\right) \div x^2$.
• Simplify the expression $\left(-4 y^4 - 2 y^3 - y^2\right) \div y^2$.

Additional Resources

• For more practice problems, visit the Khan Academy website.
• For more information on polynomial division, visit the Mathway website.

Final Answer

The final answer is D) $-9 c - 5 - \frac{1}{c}$.
Simplify: $\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3$ Q&A

Frequently Asked Questions

We've received many questions about the problem $\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3$. Here are some of the most frequently asked questions and their answers.

Q: What is the rule for dividing like bases?

A: When dividing like bases, we subtract the exponents. In this case, we have:

$$\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3 = -9 c^{4-3} - 5 c^{3-3} - c^{2-3}$$

Q: How do I simplify the expression?

A: To simplify the expression, we need to apply the rules of arithmetic operations. We have:

$$-9 c^1 - 5 c^0 - c^{-1}$$

Since $c^0 = 1$, we can simplify the expression as:

$$-9 c - 5 - \frac{1}{c}$$

Q: Why is option D) $-9 c - 5 - \frac{1}{c}$ the correct answer?

A: Option D) $-9 c - 5 - \frac{1}{c}$ is the correct answer because it matches the simplified expression we obtained:

$$-9 c - 5 - \frac{1}{c}$$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to apply the rules of exponents
• Not simplifying the expression fully
• Not checking the answer against the original expression

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems, such as:

• Simplify the expression $\left(2 x^3 + 3 x^2 + x\right) \div x^2$
• Simplify the expression $\left(-4 y^4 - 2 y^3 - y^2\right) \div y^2$

Q: Where can I find more information on polynomial division?

A: You can find more information on polynomial division at the following websites:

• Khan Academy: www.khanacademy.org
• Mathway: www.mathway.com

Conclusion

We hope this Q&A article has been helpful in answering your questions about the problem $\left(-9 c^4 - 5 c^3 - c^2\right) \div c^3$. Remember to always apply the rules of exponents and arithmetic operations when simplifying expressions, and to check your answer against the original expression.

Additional Resources

• For more practice problems, visit the Khan Academy website.
• For more information on polynomial division, visit the Mathway website.

Final Answer

The final answer is D) $-9 c - 5 - \frac{1}{c}$.