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Chapter 9: Problem 26

For Exercises 25 to \(32,\) add. Use a vertical format. $$\left(3 x^{2}+9 x\right)+(6 x-24)$$

Short Answer

Expert verified
\(3x^2 + 15x - 24\)

Step by step solution

01

Arrange the Polynomials Vertically

First, arrange the polynomials in the vertical format based on the powers of x. It's better to start from the highest power. Also, plus sign between brackets implies that we are adding together, not subtracting. So, rewriting the equation, we have: \[ \begin{{array}}{{ll}} & 3x^2 + 9x \ + & 6x - 24 \ \hline \end{{array}} \]
02

Add Like Terms

Next add like terms vertically. This is done by adding the coefficients of the terms which have the same power of x. Do this for each power of x in the equation: \[ \begin{{array}}{{ll}} & 3x^2 + 9x \ + & 6x - 24 \ \hline & 3x^2 + 15x - 24 \end{{array}} \]
03

Present the Final Answer

Simplify the given expression by writing down the terms obtained after vertical addition, which gives the final answer as follows: \(3x^2 + 15x - 24\).

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Key Concepts

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

Vertical Format
When adding polynomials, using a vertical format can simplify your calculations. In this format, each polynomial is written one below the other, aligning terms with the same degree. This setup helps to keep track of like terms and prevents common mistakes.
To illustrate, if you have polynomials like \(3x^2 + 9x\) and \(6x - 24\), you would write them as:
  • 3\(x^2\)
  • + 9\(x\)
  • + 6\(x\)
  • - 24
Then, draw a line under them to separate your work area where you will write the resulting polynomial.
Like Terms
Like terms are terms within polynomials that have the same variable raised to the same power. Identifying these terms is crucial when performing operations like addition. They need to be combined to simplify the expression effectively.
For example, in \(3x^2 + 9x\) and \(6x - 24\), \(9x\) and \(6x\) are both like terms because they contain the variable \(x\) raised to the same power of 1. Recognizing and grouping these terms makes the process of polynomial addition straightforward. By visually aligning them in a vertical format, the task becomes even simpler, especially with complex polynomials.
Coefficient Addition
When adding polynomials, the key step is to add the coefficients of like terms. Coefficients are the numbers in front of variables. It involves simple arithmetic addition.
For the expression \(9x + 6x\), you only need to add the numbers 9 and 6, which gives 15. So, the sum becomes \(15x\).
Remember:
  • Keep the variable and the power unchanged during this step.
  • Focus solely on the numbers in front.
This rule eases the procedure and helps avoid the mistake of altering the variable or its power during addition.
Simplification of Expressions
The final step in polynomial addition is simplification. Once all like terms have been added together, you receive a new expression that is easier to interpret and work with.
For the given example, after you add \(3x^2\), \(15x\), and \(-24\), your simplified polynomial is \(3x^2 + 15x - 24\).
Simplification ensures that expressions are as concise as possible, reducing errors in further calculations. This step is often signaled by writing out the terms cleanly, in standard polynomial form, typically ordered from highest to lowest power.

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