Problem 20 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ \frac{6}{7} y^{2}+\frac{1}{2} y-\frac{2}{3} y^{2}+\frac{1}{5} y $$

Step-by-Step Solution

Answer

The simplified polynomial is \(\frac{4}{21} y^{2} + \frac{7}{10} y\).

Step 1: Identify Like Terms

In the given polynomial \(\frac{6}{7} y^{2} + \frac{1}{2} y - \frac{2}{3} y^{2} + \frac{1}{5} y\), identify like terms. These are terms with the same variable and exponent. Thus, we have two sets of like terms: \(\frac{6}{7} y^{2}\) and \(-\frac{2}{3} y^{2}\); \(\frac{1}{2} y\) and \(\frac{1}{5} y\).

Step 2: Combine Like Terms for Quadratic Terms

Add the like terms \(\frac{6}{7} y^{2}\) and \(-\frac{2}{3} y^{2}\). To do this, find a common denominator, which is 21 in this case: \(\frac{6 \times 3}{21} y^{2} - \frac{2 \times 7}{21} y^{2}\). This simplifies to \(\frac{18}{21} y^{2} - \frac{14}{21} y^{2} = \frac{4}{21} y^{2}\).

Step 3: Combine Like Terms for Linear Terms

Add the like terms \(\frac{1}{2} y\) and \(\frac{1}{5} y\). Again, find a common denominator, which is 10: \(\frac{1 \times 5}{10} y + \frac{1 \times 2}{10} y\). Simplifying gives \(\frac{5}{10} y + \frac{2}{10} y = \frac{7}{10} y\).

Step 4: Write the Simplified Polynomial

Combine the results from Step 2 and Step 3 into the final simplified polynomial: \(\frac{4}{21} y^{2} + \frac{7}{10} y\).

Key Concepts

Understanding Like Terms

The concept of like terms is essential when simplifying polynomials. Like terms are terms that have the same variable raised to the same power. They can be combined through addition or subtraction. For example, in the polynomial given,

• \( \frac{6}{7} y^2 \) and \( -\frac{2}{3} y^2 \) are like terms because both contain the variable \( y \) raised to the power of 2.
• Similarly, \( \frac{1}{2} y \) and \( \frac{1}{5} y \) are also like terms since they involve the variable \( y \) raised to the power of 1.

Identifying like terms allows us to simplify expressions by combining these terms, thus making calculations easier and more straightforward.

Finding a Common Denominator

When dealing with fractions in like terms, a common denominator is necessary for addition or subtraction. This process involves finding a shared number that all denominators can divide into. Once a common denominator is found, each fraction is rewritten with this denominator, allowing the terms to be combined.
For instance, consider the quadratic terms \( \frac{6}{7} y^2 \) and \( -\frac{2}{3} y^2 \). The least common denominator here is 21.

• Convert these fractions: \( \frac{6 \times 3}{21} = \frac{18}{21} \)
• \( \frac{2 \times 7}{21} = \frac{14}{21} \).

This conversion enables straightforward subtraction, resulting in \( \frac{4}{21} y^2 \). The same process is applied to the linear terms, using 10 as their common denominator.

Combining Polynomials

Combining polynomials is the final step in simplification, where all like terms are merged into a single, simplified expression. After identifying like terms and ensuring they share a common denominator, these terms can be added or subtracted together.After simplifying the like terms separately for quadratic and linear components in our example, we can now combine them into one polynomial expression.
Here, the terms \( \frac{4}{21} y^2 \) from the quadratic simplification and \( \frac{7}{10} y \) from the linear terms are brought together to form the final polynomial:

• \( \frac{4}{21} y^2 + \frac{7}{10} y \).

This concise expression represents the simplified form of the original polynomial, ready for further use or evaluation.