Problem 97 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{w^{2}-3}{3 w^{3}+81}-\frac{2}{6 w+18}-\frac{w-4}{w^{2}-3 w+9} $$

Step-by-Step Solution

Answer

0

Step 1 - Factor Denominators

Identify and factorize the denominators. Notice that the second denominator can be factored as follows:- \(3w^3 + 81 = 3(w^3 + 27) = 3(w+3)(w^2 - 3w + 9)\)- \(6w + 18 = 6(w + 3)\)- \(w^2 - 3w + 9 = \text{already in simplest form}\)

Step 2 - Simplify the Expression

Apply the factored denominators and rewrite the expression with common denominators:\[ \frac{w^2 - 3}{3(w+3)(w^2 - 3w + 9)} - \frac{2}{6(w+3)} - \frac{w-4}{w^2 - 3w + 9} \]

Step 3 - Find Common Denominator

Find the least common denominator (LCD) for all fractions. The LCD is \(6(w+3)(w^2 - 3w + 9)\).Rewrite each fraction over this common denominator:\[ \frac{2(w^2 - 3)}{6(w+3)(w^2 - 3w + 9)} - \frac{6(w-4)}{6(w+3)(w^2 - 3w + 9)} - \frac{2(w^2 - 3w+9)}{6(w+3)(w^2 - 3w + 9)}\]

Step 4 - Combine Fractions

Now that the denominators are the same, combine the numerators:\[\frac{2(w^2 - 3) - 6(w-4) - 2(w^2 - 3w + 9)}{6(w+3)(w^2 - 3w + 9)}\]

Step 5 - Simplify Numerator

Simplify the combined numerator:\[\frac{2w^2 - 6 - 6w + 24 - 2w^2 + 6w - 18}{6(w+3)(w^2 - 3w + 9)} = \frac{2w^2 - 2w^2 - 6 + 24 - 18 + 6w - 6w}{6(w+3)(w^2 - 3w + 9)} = \frac{0}{6(w+3)(w^2 - 3w + 9)}\]

Step 6 - Final Simplification

Since the numerator is zero, the entire expression simplifies to zero:\[0\]

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential step in simplifying rational expressions. When you factor a polynomial, you are breaking it down into simpler polynomials that multiply together to give you the original polynomial. This helps in identifying common factors and making it easier to simplify expressions later on.
 
In the given exercise, we identified and factored the denominators:
- For the first fraction:\[3w^3 + 81 = 3(w^3 + 27) = 3(w + 3)(w^2 - 3w + 9)\]
- For the second fraction: \[6w + 18 = 6(w + 3)\]
- The third fraction's denominator \(w^2 - 3w + 9\) remains unchanged since it is already in its simplest form. Factoring polynomials like these prepares the expression for the next step: finding a common denominator.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when adding or subtracting fractions. The LCD is the smallest expression that all the denominators can divide into. This allows you to combine the fractions easily.
 
In the provided exercise, the factored forms of the denominators were:
- \(3(w + 3)(w^2 - 3w + 9)\)
- \(6(w + 3)\)
- \((w^2 - 3w + 9)\)
By examining these, we see that the common terms are \((w + 3)\) and \((w^2 - 3w + 9)\). Including the largest coefficient, the LCD is given by:
\[6(w + 3)(w^2 - 3w + 9)\]

Combining Fractions

Once you have a common denominator, you can rewrite each fraction with this denominator and combine them.
 
From the exercise:
- The first fraction was rewritten as: \[ \frac{2(w^2 - 3)}{6(w+3)(w^2 - 3w + 9)}\]
- The second fraction was rewritten as: \[ \frac{6(w-4)}{6(w+3)(w^2 - 3w + 9)}\]
- The third fraction was rewritten as: \[ \frac{2(w^2 - 3w + 9)}{6(w+3)(w^2 - 3w + 9)}\]
These fractions are now ready to be combined because they share a common denominator. Therefore, the numerators are added or subtracted while keeping the common denominator the same:

Simplifying Expressions

After combining the fractions, simplify the resulting expression. This involves combining like terms and reducing the expression to its simplest form. In this exercise:
 
The combined fraction is:
\[\frac{2(w^2 - 3) - 6(w-4) - 2(w^2 - 3w + 9)}{6(w+3)(w^2 - 3w + 9)}\]
Expanding and combining like terms gives us:
\[2w^2 - 6 - 6w + 24 - 2w^2 + 6w - 18\]
Simplifying these terms:
\[2w^2 - 2w^2 - 6 + 24 - 18 + 6w - 6w = 0\]
Since the numerator becomes zero, the entire expression simplifies to zero:
\[ \frac{0}{6(w+3)(w^2 - 3w + 9)} = 0 \]
This fully simplifies the original problem.