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Chapter 3: Problem 9

Divide. $$-40 n^{9} \div(-2 n)$$

Short Answer

Expert verified
Answer: The simplified division is $$20n^8$$.

Step by step solution

01

Divide the coefficients

Divide -40 by -2. A negative number divided by a negative number results in a positive number. So, the division will result in 20.
02

Divide the variables

Now, let's divide the variables with the exponents. We have $$n^9 \div n$$. According to the exponent rules, when dividing two terms with the same base and exponents, we subtract the exponents. Here, we have $$\frac{n^9}{n^1}$$ which results in $$n^{(9-1)}$$, which simplifies to $$n^8$$.
03

Combine the results

Combine the results from Step 1 and Step 2: $$20n^8$$. So, the simplified division of $$-40 n^9 \div(-2 n)$$ is $$20n^8$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient Division
Dividing coefficients involves handling the numerical parts of the terms in the expression. In the exercise, you need to divide -40 by -2. Here's a helpful tip:
  • When you divide a negative number by another negative, the result is positive. Think of it like cancelling out the negativity.
  • In our example, dividing -40 by -2 gives you 20, turning the negative division into a positive outcome.
This essential step prepares you by simplifying the numbers before moving onto the variables and their exponents.
Exponent Rules
When working with expressions involving exponents, understanding exponent rules is crucial. The step in the exercise involves dividing the variable parts of the terms, like this: \(n^9 \div n^1\). Here are some key points:
  • The main rule is that when dividing two terms with the same base, you subtract the exponents: \(n^{9-1}\).
  • This results in \(n^8\), simplifying the expression and revealing the new power of the variable.
Remember, these exponent rules help reduce complex polynomial divisions into manageable steps, making calculations much easier.
Negative Number Division
Handling negative numbers can often be tricky, but there's a straightforward rule: dividing a negative by a negative results in a positive. In our division of -40 by -2, this rule is applied:
  • Both numbers are negative, thus their division results in a positive number, which is consistent across all situations.
  • This Principle not only applies to coefficients but also prepares you for more complex functions involving negative numbers.
Mastering this rule helps make polynomial division, such as this exercise, smoother and less error-prone.
Simplifying Expressions
The final part of tackling polynomial division is simplifying the expression. After separately dividing the coefficients and variables, it鈥檚 time to combine your results:
  • First, you have the coefficient result, which was 20 after dividing -40 by -2.
  • Next, from the exponent division, you ended up with \(n^8\).
  • Combine them to form the simplified expression: \(20n^8\).
This synthesis is key to ensuring that your final expression is as simple and clear as possible, allowing for easy interpretation and usage in more complex problems.

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