Problem 53 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(\frac{3}{8} s^{8}-\frac{3}{4} s^{7}\right)-\left(\frac{1}{3} s^{8}+\frac{1}{5} s^{7}\right) $$

Step-by-Step Solution

Answer

The result of the subtraction is \( \frac{1}{24} s^8 - \frac{19}{20} s^7 \).

Step 1: Distribute the Negative Sign

We need to distribute the negative sign across the polynomial being subtracted, \( \left(\frac{1}{3} s^8 + \frac{1}{5} s^7\right) \). This changes the expression to \[ \frac{3}{8} s^8 - \frac{3}{4} s^7 - \left( \frac{1}{3} s^8 + \frac{1}{5} s^7 \right) = \frac{3}{8} s^8 - \frac{3}{4} s^7 - \frac{1}{3} s^8 - \frac{1}{5} s^7 \].

Step 2: Combine Like Terms

Combine the terms with the same degree from the distributed expression. We start by combining the \( s^8 \) terms and the \( s^7 \) terms:1. \( s^8 \): \[ \frac{3}{8} s^8 - \frac{1}{3} s^8 = \left( \frac{3}{8} - \frac{1}{3} \right) s^8 \]2. \( s^7 \): \[ -\frac{3}{4} s^7 - \frac{1}{5} s^7 = -\left( \frac{3}{4} + \frac{1}{5} \right) s^7 \].

Step 3: Find a Common Denominator

To simplify the expression, we need common denominators.1. For \( \frac{3}{8} - \frac{1}{3} \), the common denominator is 24. \[ \frac{3}{8} = \frac{9}{24}, \quad \frac{1}{3} = \frac{8}{24} \].2. For \( \frac{3}{4} + \frac{1}{5} \), the common denominator is 20. \[ \frac{3}{4} = \frac{15}{20}, \quad \frac{1}{5} = \frac{4}{20} \].

Step 4: Simplifying the Expression

Replace with common denominators and simplify:1. For \( s^8 \): \[ \frac{9}{24} - \frac{8}{24} = \frac{1}{24} \Rightarrow \frac{1}{24} s^8 \].2. For \( s^7 \): \[ -\frac{15}{20} - \frac{4}{20} = -\frac{19}{20} \Rightarrow -\frac{19}{20} s^7 \].Thus, the final expression is \[ \frac{1}{24} s^8 - \frac{19}{20} s^7 \].

Key Concepts

Combining like terms

In polynomial subtraction, particularly when dealing with expressions such as \( \left(\frac{3}{8} s^{8}-\frac{3}{4} s^{7}\right)-\left(\frac{1}{3} s^{8}+\frac{1}{5} s^{7}\right) \), a key step is combining like terms. Like terms are those that have the same variable raised to the same power. This means they look identical except for their coefficients.

• Consider the terms \( \frac{3}{8} s^8 \) and \( \frac{1}{3} s^8 \). These are like terms because they both have \( s^8 \).
• Similarly, \( -\frac{3}{4} s^7 \) and \( \frac{1}{5} s^7 \) are like terms because they involve \( s^7 \).

Combining like terms involves adding or subtracting these terms. You only work with the coefficients, and the common variable doesn't change. This step simplifies the polynomial and helps in further calculations. Simply perform the necessary addition or subtraction on the coefficients to simplify.

Common denominator

In order to successfully combine like terms, fractions involved must share a common denominator. A common denominator is a shared multiple of the denominators of two fractions, allowing for straightforward addition or subtraction.

• For the terms \( \frac{3}{8} s^8 \) and \( \frac{1}{3} s^8 \), the denominators are 8 and 3. The least common multiple is 24, meaning we convert both fractions to have the denominator 24. Convert \( \frac{3}{8} \) to \( \frac{9}{24} \) and \( \frac{1}{3} \) to \( \frac{8}{24} \).
• For \( -\frac{3}{4} s^7 \) and \( \frac{1}{5} s^7 \), the denominators 4 and 5 need a common denominator of 20. Thus, \( \frac{3}{4} \) becomes \( \frac{15}{20} \), and \( \frac{1}{5} \) becomes \( \frac{4}{20} \).

With this method, subtraction or addition becomes direct, simplifying the task and helping ensure accuracy in the solution.

Distributing negative sign

A crucial step in subtracting polynomials like \( \left(\frac{3}{8} s^{8}-\frac{3}{4} s^{7}\right)-\left(\frac{1}{3} s^{8}+\frac{1}{5} s^{7}\right) \) involves distributing the negative sign. When you subtract one polynomial from another, distributing the negative sign changes the sign of each term in the polynomial that follows the minus sign.

• The expression \( \left(\frac{1}{3} s^8 + \frac{1}{5} s^7 \) becomes \( -\frac{1}{3} s^8 - \frac{1}{5} s^7 \) when the negative is distributed.
• This step ensures that every part of the polynomial is correctly subtracted, preventing common errors.

By carefully managing signs, we ensure the integrity of the operation, allowing accurate simplification and results. This lays down the groundwork for combining like terms.