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Chapter 13: Problem 41

Find each sum or difference. $$\left(13 n^{2}+6 n+5\right)-\left(6 n^{2}+5\right)$$

Short Answer

Expert verified
The simplified expression is \(7n^2 + 6n\).

Step by step solution

01

Identify the Expression

The problem requires simplifying the expression \( (13n^2 + 6n + 5) - (6n^2 + 5) \). This is a subtraction between two polynomials.
02

Distribute the Negative Sign

Apply the negative sign in front of the second polynomial to each term inside the bracket: \( 13n^2 + 6n + 5 - 6n^2 - 5 \).
03

Combine Like Terms

Group and combine the like terms. Start with the \(n^2\) terms: \(13n^2 - 6n^2 = 7n^2\). Then handle the constant terms: \(5 - 5 = 0\). Finally, handle any remaining terms: \(6n\).
04

Present the Simplified Expression

After combining the terms, the expression simplifies to \(7n^2 + 6n\).

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Key Concepts

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

Subtraction of Polynomials
When tasked with subtracting polynomials, the key is to treat each polynomial like a separate mathematical entity. This particular exercise posed the question of simplifying \( \left(13n^2 + 6n + 5\right) - \left(6n^2 + 5\right) \). The challenge is in handling the subtraction of one polynomial from another. Here's a simple, friendly breakdown:
  • Start by writing out both polynomials, explicitly showing their terms.
  • Subtract the second polynomial from the first by changing each term's sign in the second polynomial.
Remember, this subtraction acts as a way of redistributing and repositioning values, letting you eventually simplify it effectively. Our ultimate goal is to achieve fewer terms and a more concise expression.
Combining Like Terms
Combining like terms is a fundamental concept that helps reduce polynomials into their simplest form. In the simplification process, you identify terms that are similar in composition, like powers of \(n\) and constants. For our exercise, we had terms such as:\[ 13n^2 + 6n + 5 - 6n^2 - 5 \]The objective was to recognize and combine terms as follows:
  • \(n^2\) terms: Add \(13n^2\) and \(-6n^2\) together to get \(7n^2\).
  • Constant terms: Subtract \(5\) from \(5\), resulting in \(0\), which we can essentially remove from the expression.
  • Finally, bring down the \(6n\) which is unaffected by the constant term subtraction.
Noticing these common threads allows for easier manipulation and combination, thus simplifying the overall expression to \(7n^2 + 6n\).
Distributing Negative Sign
Handling the negative sign correctly is crucial when subtracting polynomials. It might seem a bit tricky initially, but with practice, it becomes second nature. Let鈥檚 reconsider our expression:\( (13n^2 + 6n + 5) - (6n^2 + 5) \).The negative sign before the parenthesis requires that we:
  • "Distribute" it to each term within the second polynomial.
  • This changes the expression to \( 13n^2 + 6n + 5 - 6n^2 - 5 \).
By doing so, each term within the parenthesis flips its sign. Ensuring you handle this step accurately paves the way to correct simplification as you go forward. Performing these steps without error sets the stage for combining like terms effortlessly.

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