Problem 54 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(\frac{5}{6} q^{9}-\frac{4}{5} q^{8}\right)-\left(\frac{1}{4} q^{9}+\frac{3}{8} q^{8}\right) $$

Step-by-Step Solution

Answer

\(\frac{7}{12} q^{9} - \frac{47}{40} q^{8}\)

Step 1: Rewrite the Expression

Start by rewriting the given expression, breaking it down explicitly into its components. The original expression is: \[\left(\frac{5}{6} q^{9} - \frac{4}{5} q^{8}\right) - \left(\frac{1}{4} q^{9} + \frac{3}{8} q^{8}\right)\] Remember that when you subtract polynomials, you distribute the negative sign into the second set of parentheses.

Step 2: Distribute the Negative Sign

Distribute the negative sign across the terms in the second polynomial. This changes the sign of each term inside the parentheses:\[\frac{5}{6} q^{9} - \frac{4}{5} q^{8} - \frac{1}{4} q^{9} - \frac{3}{8} q^{8}\]

Step 3: Combine Like Terms

Add together the like terms. These are the terms that contain the same power of \(q\):1. For \(q^{9}\) terms: \[\frac{5}{6} q^{9} - \frac{1}{4} q^{9}\]2. For \(q^{8}\) terms:\[-\frac{4}{5} q^{8} - \frac{3}{8} q^{8}\]

Step 4: Calculate the Coefficients for q^{9}

First, find a common denominator for the fractions \(\frac{5}{6}\) and \(\frac{1}{4}\) to combine them. The least common denominator is 12:\[\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}\]Subtract the \(q^{9}\) terms:\[\frac{10}{12} q^{9} - \frac{3}{12} q^{9} = \frac{7}{12} q^{9}\]

Step 5: Calculate the Coefficients for q^{8}

Find a common denominator for the fractions \(\frac{4}{5}\) and \(\frac{3}{8}\). The least common denominator is 40:\[\frac{4}{5} = \frac{32}{40}, \quad \frac{3}{8} = \frac{15}{40}\]Subtract the \(q^{8}\) terms:\[-\frac{32}{40} q^{8} - \frac{15}{40} q^{8} = -\frac{47}{40} q^{8}\]

Step 6: Write the Final Expression

Combine the results from the previous two steps to write the final result of the subtraction:\[\frac{7}{12} q^{9} - \frac{47}{40} q^{8}\]

Key Concepts

Distributing the Negative Sign

When subtracting polynomials, a crucial step is distributing the negative sign from the subtraction across the terms in the second polynomial. This is similar to multiplying each term by -1.

Think of it like removing a wrapper. The negative sign will change the sign of each term inside the parentheses:

• Positive terms become negative
• Negative terms become positive

For example, in the expression \((x - (y + z))\), distribute the negative: \(-y - z\).

In our polynomial, \(\frac{1}{4} q^{9} + \frac{3}{8} q^{8}\) becomes \(- \frac{1}{4} q^{9} - \frac{3}{8} q^{8}\). This step ensures that the signs are correct when combining terms.

Combining Like Terms

Once the negative sign is distributed, look for like terms in the expression. Like terms are terms that contain the same variable raised to the same power.

Here, the terms \(\frac{5}{6} q^{9}\) and \(- \frac{1}{4} q^{9}\) both contain \(q^{9}\). Likewise, \(- \frac{4}{5} q^{8}\) and \(- \frac{3}{8} q^{8}\) both contain \(q^{8}\).

• Combine the coefficients of each set of like terms separately.
• Add or subtract the fractions as needed.

This results in a simplified expression where you have one term for each power of the variable. In our example, combining like terms simplifies the expression to \(\frac{7}{12} q^{9} - \frac{47}{40} q^{8}\).

Finding Common Denominators

To accurately combine fractions, we must find a common denominator. This is essential when adding or subtracting coefficients, particularly if they are fractions.

Imagine we have different pieces of a circle divided into different numbers of segments. Adjust them to have the same total number of segments so they can be added or subtracted easily.

Here's how to find common denominators:

• Identify the least common denominator (LCD) between the fractions.
• Convert each fraction to an equivalent fraction with the LCD.
• Add or subtract as appropriate.

For instance, to subtract \(\frac{5}{6}\) and \(\frac{1}{4}\), convert them both to twelfths: \(\frac{5}{6} = \frac{10}{12}\) and \(\frac{1}{4} = \frac{3}{12}\).

This ensures that calculations involving fractions are precise, allowing the polynomial terms to be accurately combined.