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Chapter 5: Problem 90

Add. $$ (9 a b-3 a c+5 b c)+(13 a b-15 a c-8 b c) $$

Short Answer

Expert verified
The sum is \( 22ab - 18ac - 3bc \).

Step by step solution

01

- Identify Like Terms

First, list the like terms from each polynomial. Like terms have the same variables raised to the same powers. Here, the like terms are: 1. Terms with `ab`: `9ab` and `13ab`2. Terms with `ac`: `-3ac` and `-15ac`3. Terms with `bc`: `5bc` and `-8bc`
02

- Add Like Terms

Next, add the coefficients of the like terms:1. For the `ab` terms: \[ 9ab + 13ab = (9 + 13)ab = 22ab \]2. For the `ac` terms: \[ -3ac - 15ac = (-3 - 15)ac = -18ac \]3. For the `bc` terms: \[ 5bc - 8bc = (5 - 8)bc = -3bc \]
03

- Write the Final Sum

Combine the results from Step 2 into a single expression:\( 22ab - 18ac - 3bc \)

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Key Concepts

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

Understanding Like Terms
In polynomials, 'like terms' are terms that have the same variables raised to the same powers.
This means that for terms to be considered 'like', they must match exactly in terms of their variable parts.
For example:
  • In the expression (9ab - 3ac + 5bc) + (13ab - 15ac - 8bc), we have several terms.
  • The like terms here can be identified as:
    Terms with `ab`: `9ab` and `13ab`
    Terms with `ac`: `-3ac` and `-15ac`
    Terms with `bc`: `5bc` and `-8bc`
Identifying like terms is the first step in simplifying polynomial expressions.
When like terms are identified correctly, it becomes easier to add or subtract the coefficients.
What Are Coefficients?
Coefficients are the numerical factors in terms of a polynomial.
They are the numbers that multiply the variables.
For example:
  • In the term `9ab`, the coefficient is 9.
  • In the term `-3ac`, the coefficient is -3.
  • In the term `5bc`, the coefficient is 5.
Coefficients play a crucial role in polynomial addition because we focus on them when combining like terms.
Always match terms with the exact same variable part and then add or subtract the coefficients.
For instance, in the given example:
  • `9ab` and `13ab` are like terms, so their coefficients 9 and 13 can be added: 9 + 13 = 22, resulting in `22ab`.
  • Similarly, `-3ac` and `-15ac` are like terms, so their coefficients -3 and -15 can be added: -3 + -15 = -18, resulting in `-18ac`.
This step is all about handling the numbers attached to the variables.
Combining Polynomials
Combining polynomials involves adding or subtracting the terms of the polynomials.
The key is to correctly align and combine the like terms.
Here's how it works step-by-step:
1. **Identify Like Terms**: Match terms with the same variables raised to the same powers.
2. **Add/Subtract Coefficients**: Once like terms are identified, add or subtract their coefficients.
  • For example, in the problem, you start by identifying `9ab` and `13ab` as like terms.
  • Add their coefficients: 9 + 13, resulting in `22ab`.
3. **Combine the Results**: After performing addition or subtraction on like terms, combine the new terms into a single polynomial.
  • From the given problem, the final combined polynomial is:
    `22ab - 18ac - 3bc`.
Combining polynomials can also involve using subtraction, so it's important to keep careful track of positive and negative signs when working with coefficients.

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