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Chapter 6: Problem 23

Simplify. $$ \left(9 r^{2}+6 r+16\right)-\left(8 r^{2}+7 r+10\right) $$

Short Answer

Expert verified
The simplified expression is \(r^{2} - r + 6\).

Step by step solution

01

Distribute the Negative Sign

The expression is \( \left(9r^{2} + 6r + 16\right) - (8r^{2} + 7r + 10) \). To simplify, first distribute the negative sign across the second set of parentheses: \( 9r^{2} + 6r + 16 - 8r^{2} - 7r - 10 \).
02

Combine Like Terms

Now, gather the like terms together: \((9r^{2} - 8r^{2})\), \((6r - 7r)\), and \((16 - 10)\).
03

Simplify Each Group

Simplify each group of like terms: \(9r^{2} - 8r^{2} = r^{2}\), \(6r - 7r = -r\), and \(16 - 10 = 6\).
04

Write Final Expression

Combine all simplified terms to get the final simplified expression: \(r^{2} - r + 6\).

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Key Concepts

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

Simplifying Polynomials
When you start simplifying polynomials, you aim to make the expression as compact and simple as possible. Polynomial expressions are mathematical statements involving variables and coefficients, using operations of addition, subtraction, and multiplication. Simplifying is all about making these expressions easier to understand and work with.

To simplify a polynomial, follow these general steps:
  • Combine like terms, which are terms that have the same variable raised to the same power.
  • Perform operations such as addition or subtraction between these like terms.
  • Make sure to account for all parts of the expression, whether they include variables or are constant terms.
By simplifying, you convert a complex equation into something more straightforward and manageable, which is important for solving mathematical problems effectively.
Distributing Negative Signs
Distributing a negative sign is a pivotal step when simplifying expressions, especially in polynomial subtraction. When you see a subtraction sign in front of a set of parentheses, it means you need to change the sign of every term inside the parentheses.

Consider this example:
  • The polynomial expression is \((9r^{2} + 6r + 16) - (8r^{2} + 7r + 10)\).
  • To distribute the negative sign, you transform it to \(9r^{2} + 6r + 16 - 8r^{2} - 7r - 10\).
Doing so changes the signs of all terms from \(8r^{2} + 7r + 10\) to \(-8r^{2} - 7r - 10\).

This step ensures you maintain mathematical accuracy by correctly representing the subtraction operation.
Combining Like Terms
Once you've distributed any negative signs, the next move is combining like terms. Like terms are terms that have the same variable parts raised to the same power. In polynomial simplification, combining these terms is essential.

Here's how you achieve that:
  • Identify like terms, such as those with \(r^{2}\), \(r\), or constants.
  • Perform simple arithmetic operations to combine them.
  • From our example: combine \((9r^{2} - 8r^{2})\), \((6r - 7r)\), and \((16 - 10)\).
  • This results in \(r^{2}\), \(-r\), and \(+6\).
Combining like terms effectively reduces the polynomial to its simplest form, making it easier to work with in further calculations.

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