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Chapter 5: Problem 114

Simplify. \(0.1 n^{2}+5-\left(-0.3 n^{2}+n-6\right)\)

Short Answer

Expert verified
The simplified expression is \[0.4 n^{2} - n + 11\].

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign across the terms inside the parentheses. This changes the signs of each term inside: \[0.1 n^{2} + 5 - (-0.3 n^{2} + n - 6) = 0.1 n^{2} + 5 + 0.3 n^{2} - n + 6\]
02

Combine Like Terms

Next, combine the like terms. Group the terms with the same variable together: \[ (0.1 n^{2} + 0.3 n^{2}) - n + (5 + 6)\] Simplify each group: \[0.4 n^{2} - n + 11\]

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Key Concepts

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

Distributive Property
The distributive property is a core technique in algebra that helps in simplifying expressions by distributing or multiplying a term across others inside parentheses. In our problem, we need to distribute the negative sign over the terms inside the parentheses. This means we change the sign of each term inside the parentheses:

Step 1 of the solution is to distribute the negative sign across the terms:
  • Original expression: [0.1n^{2} + 5 - (-0.3n^{2} + n -6)]
  • Change the signs: [0.1n^{2} + 5 + 0.3n^{2} - n + 6]

Distributing the negative sign ensures that we can work with positive terms and avoid mistakes.
Combining Like Terms
Combining like terms is another essential algebraic process. Like terms contain the same variables raised to the same power. Simplifying an expression requires grouping and adding or subtracting these terms. In our exercise, we identified like terms and grouped them:

The original expression after distributing the negative sign is: [0.1n^{2} + 5 + 0.3n^{2} - n + 6]
We then combine like terms:
  • Combine the constants: [5 + 6 = 11]
  • Combine the [n^{2}] terms: [0.1n^{2} + 0.3n^{2} = 0.4n^{2}]
  • The [n] term remains as [-n]

Thus, the simplified expression is: [0.4n^{2} - n + 11]
Polynomial Expressions
Polynomial expressions are sums of terms consisting of variables raised to non-negative integer powers and coefficients. Each term in a polynomial is known as a monomial. Polynomials can be combined, added, and simplified using rules like distributing and combining like terms.

Our exercise involves a polynomial expression: [0.1n^{2} + 5 - (-0.3n^{2} + n - 6)]

We simplified the polynomial using the distributive property and combining like terms, resulting in: [0.4n^{2} - n + 11]

Key points for simplifying polynomial expressions:
  • Identify and distribute any negatives or constants across terms.
  • Group like terms based on their variables and exponents.
  • Combine like terms by adding or subtracting their coefficients.

By following these steps, simplifying even complex polynomial expressions becomes straightforward.

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Most popular questions from this chapter

Construct three like terms of degree 4.

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=3 x-2\\\ &g(x)=2 x-8 \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{2}{x}\\\ &g(x)=x^{2}-4 \end{aligned} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{7 x}{x-2}\\\ &g(x)=3 x+7 \end{aligned} $$

Simplify. $$ \frac{125^{-4}\left(25^{2}\right)^{4}}{125} $$
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