Problem 73 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{s+7}{s+3}-\frac{s-3}{s+7}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{14s + 58}{(s+3)(s+7)} \).

Step 1: Identify Common Denominator

In order to subtract the fractions \( \frac{s+7}{s+3} - \frac{s-3}{s+7} \), we need to have a common denominator. The common denominator of \(s+3\) and \(s+7\) is \((s+3)(s+7)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Adjust each fraction to have the common denominator \((s+3)(s+7)\):\[\frac{s+7}{s+3} \times \frac{s+7}{s+7} = \frac{(s+7)^2}{(s+3)(s+7)}\] and\[\frac{s-3}{s+7} \times \frac{s+3}{s+3} = \frac{(s-3)(s+3)}{(s+7)(s+3)}\].

Step 3: Subtract the Fractions

Subtract the two fractions that now have a common denominator:\[\frac{(s+7)^2}{(s+3)(s+7)} - \frac{(s-3)(s+3)}{(s+7)(s+3)} = \frac{(s+7)^2 - (s-3)(s+3)}{(s+3)(s+7)}.\]

Step 4: Expand and Simplify the Numerator

Expand the expressions in the numerator:\[ (s+7)^2 = s^2 + 14s + 49\] and\[(s-3)(s+3) = s^2 - 9.\]Subtract these expanded forms:\[(s^2 + 14s + 49) - (s^2 - 9) = 14s + 58.\]

Step 5: Simplify the Fraction

Now simplify the fraction:\[\frac{14s + 58}{(s+3)(s+7)}.\]Since there are no common factors that can be further canceled between the numerator and the denominator, this is the simplified form.

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential to perform operations such as addition or subtraction. The denominators must be the same so that the fractions are talking about the same size pieces. In our exercise, we have the fractions \( \frac{s+7}{s+3} \) and \( \frac{s-3}{s+7} \). Here, the denominators \(s+3\) and \(s+7\) are different. We cannot directly subtract these fractions, just like you can't subtract apples from oranges.

To find the common denominator, we multiply the denominators together. This gives us \((s+3)(s+7)\) because these are the easiest shared units that cover both fractions' denominators. Essentially, finding a common denominator lets us reframe two unlike fractions as like fractions, making operations possible.

• Make sure you understand what a denominator is: it's the bottom part of the fraction, showing how many parts the whole is divided into.
• By multiplying the unique factors, you find the smallest expression that both denominators can share.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. Once you have a common denominator and have combined the fractions, the goal is to simplify, if possible. In our exercise, after subtraction, we arrived at the fraction \( \frac{14s + 58}{(s+3)(s+7)} \). Simplifying involves canceling any common factors from the numerator and the denominator.

In this case, review both the numerator and denominator. We look for common factors; for linear polynomials like \( 14s + 58 \), factors usually are constants, gcd or identical polynomial terms that occur in the denominator. Here, since \( 14s + 58 \) doesn't share any factors with \( (s+3)(s+7) \) other than the trivial case, the fraction is already in its simplest form.

• Always factorize expressions fully to check for reducible terms.
• Simplifying makes fractions easier to interpret and work with.

Expanding Polynomials

Expanding polynomials is a method used to remove brackets in polynomial expressions by applying the distributive property. For our exercise, expanding is a necessary step for both clarity and computation.

In our example, the expressions you'll expand are \((s+7)^2\) and \((s-3)(s+3)\). To expand \((s+7)^2\), apply the formula \((a+b)^2 = a^2 + 2ab + b^2\). This results in \(s^2 + 14s + 49\). Similarly, for \((s-3)(s+3)\), use the difference of squares formula \(a^2 - b^2\) because it's in the form \((a-b)(a+b)\). This expansion results in \(s^2 - 9\).

• Expanding helps you simplify by showing all terms clearly.
• Also, it reveals like terms that can be combined and other simplification steps.