Problem 32 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(\frac{1}{16} r^{6}+\frac{1}{2} r^{3}-\frac{11}{12}\right)+\left(\frac{9}{16} r^{6}+\frac{9}{4} r^{3}+\frac{1}{12}\right) $$

Step-by-Step Solution

Answer

\(\frac{5}{8}r^6 + \frac{11}{4}r^3 - \frac{5}{6}\)

Step 1: Identify Like Terms

First, we identify the like terms in each polynomial. We have terms with \(r^6\), terms with \(r^3\), and constant terms in both polynomials.

Step 2: Add the \(r^6\) Terms

Add the coefficients of the terms with \(r^6\): \(\frac{1}{16} + \frac{9}{16} = \frac{10}{16} = \frac{5}{8}\). Thus, the \(r^6\) term of the sum is \(\frac{5}{8}r^6\).

Step 3: Add the \(r^3\) Terms

Add the coefficients of the terms with \(r^3\): \(\frac{1}{2} + \frac{9}{4} = \frac{2}{4} + \frac{9}{4} = \frac{11}{4}\). Thus, the \(r^3\) term of the sum is \(\frac{11}{4}r^3\).

Step 4: Add the Constant Terms

Add the constant terms: \(-\frac{11}{12} + \frac{1}{12} = -\frac{10}{12} = -\frac{5}{6}\). Thus, the constant of the sum is \(-\frac{5}{6}\).

Step 5: Write the Final Expression

Combine all the results: the sum of the polynomials is \(\frac{5}{8}r^6 + \frac{11}{4}r^3 - \frac{5}{6}\).

Key Concepts

Understanding Like Terms in Polynomials

When working with polynomial addition, a critical concept is identifying and combining like terms. Like terms are terms that have the same variable raised to the same power. For example, if you have \(r^6\) and \(r^6\) in different parts of a polynomial, these are considered like terms and can be combined by adding or subtracting their coefficients.
In the given exercise, you needed to identify the terms containing \(r^6\), \(r^3\), and the constant terms. Each of these categories represents like terms:

• Terms with \(r^6\)
• Terms with \(r^3\)
• Constant numbers without any variables

Recognizing these groups allows us to tackle each like term category effectively, ensuring that our polynomial addition process remains organized and accurate.

The Role of Coefficients in Polynomials

When adding polynomials, the coefficient plays a vital role. A coefficient is a number that multiplies the variable in a term. It essentially scales the variable term and determines how much of that particular term we have.
For instance, in the term \(\frac{1}{16}r^6\), \(\frac{1}{16}\) is the coefficient. When we add polynomial terms like \(\frac{1}{16}r^6\) and \(\frac{9}{16}r^6\), we only add the coefficients. This produces the new coefficient, which is \(\frac{10}{16}\), or simplified, \(\frac{5}{8}\).
Keep these points in mind:

• Only add coefficients of like terms.
• Simplify coefficients if possible—this often involves reducing fractions.

Understanding coefficients and their manipulation is essential to mastering polynomial addition.

Working with Constant Terms

Constant terms in a polynomial are the numbers without any attached variables. In our exercise, these constants are \(-\frac{11}{12}\) and \(\frac{1}{12}\). Although they might seem less significant, they must also be combined correctly.
To handle constant terms, simply add or subtract them as you would with regular numbers. In this example, adding the constants results in:

• \(-\frac{11}{12} + \frac{1}{12} = -\frac{10}{12}\)
• This simplifies to \(-\frac{5}{6}\)

It's crucial to simplify the result to its lowest terms to maintain the integrity and clarity of the final polynomial expression. Proper handling of constant terms ensures the polynomials are fully simplified and accurately represented.

Simplifying Fractions during Polynomial Addition

During polynomial addition, you will often encounter fractions, as with coefficients and constant terms. Simplifying these fractions helps express your answer in the simplest form, making it easier to understand and work with.
To simplify fractions, find the greatest common factor (GCF) of the numerator and the denominator and divide both by that number. For example, with \(\frac{10}{16}\):

• The GCF of 10 and 16 is 2, so divide both by 2.
• This simplifies \(\frac{10}{16}\) to \(\frac{5}{8}\).

Another instance is when we simplify \(-\frac{10}{12}\) to \(-\frac{5}{6}\) by dividing the numerator and the denominator by 2.
Simplifying fractions is crucial for clarity and precision in your results. It ensures that your final polynomial is easily interpretable and correct.