Problem 101 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{w^{2}+3}{w^{3}-8}-\frac{2 w}{w^{2}-4} $$

Step-by-Step Solution

Answer

The result is \( \frac{-w^3 - 2w^2 - 5w + 6}{(w - 2)(w^2 + 2w + 4)(w + 2)} \)

Step 1 - Factor the denominators

Examine the denominators in each fraction. The first denominator is \(w^3 - 8\), which can be factored using the difference of cubes: \(w^3 - 8 = (w - 2)(w^2 + 2w + 4)\). The second denominator is \(w^2 - 4\), which can be factored using the difference of squares: \(w^2 - 4 = (w - 2)(w + 2)\).

Step 2 - Find the common denominator

The least common denominator (LCD) is the product of all distinct factors: \((w - 2)(w^2 + 2w + 4)(w + 2)\).

Step 3 - Rewrite each fraction with the common denominator

Rewrite each fraction with the LCD as the new denominator. For the first fraction: \[ \frac{w^2 + 3}{(w - 2)(w^2 + 2w + 4)} \times \frac{(w + 2)}{(w + 2)} = \frac{(w^2 + 3)(w + 2)}{(w - 2)(w^2 + 2w + 4)(w + 2)} \] For the second fraction: \[ \frac{2w}{(w - 2)(w + 2)} \times \frac{(w^2 + 2w + 4)}{(w^2 + 2w + 4)} = \frac{2w(w^2 + 2w + 4)}{(w - 2)(w^2 + 2w + 4)(w + 2)} \]

Step 4 - Combine the fractions

Subtract the numerators while keeping the common denominator: \[ \frac{(w^2 + 3)(w + 2) - 2w(w^2 + 2w + 4)}{(w - 2)(w^2 + 2w + 4)(w + 2)} \] Simplify the numerator.

Step 5 - Expand and simplify the numerator

Expand and simplify the numerator: \[ (w^2 + 3)(w + 2) = w^3 + 2w^2 + 3w + 6 \] \[ 2w(w^2 + 2w + 4) = 2w^3 + 4w^2 + 8w \] Now subtract the simplified terms: \[ w^3 + 2w^2 + 3w + 6 - 2w^3 - 4w^2 - 8w = -w^3 - 2w^2 - 5w + 6 \] So, the combined fraction is: \[ \frac{-w^3 - 2w^2 - 5w + 6}{(w - 2)(w^2 + 2w + 4)(w + 2)} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a key algebraic process that involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This can aid in simplifying expressions and solving equations.
In the given exercise, we've factored the denominators to make addition or subtraction easier. For example, the polynomial \(w^3 - 8\) is a difference of cubes and factors into \((w - 2)(w^2 + 2w + 4)\). Similarly, \(w^2 - 4\) is a difference of squares and factors into \((w - 2)(w + 2)\).
Recognizing these patterns is crucial. The formulas to remember are:

• Difference of Cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)

Master these patterns to factor polynomials effectively!

Common Denominators

Finding a common denominator is essential when adding or subtracting fractions. The common denominator is the least common multiple of the original denominators. This allows fractions to be rewritten with the same denominator, making operations straightforward.
In our example, after factoring the denominators \(w^3 - 8\) and \(w^2 - 4\), we found the least common denominator (LCD) to be \((w - 2)(w^2 + 2w + 4)(w + 2)\).
To rewrite each fraction with this common denominator, multiply the numerator and denominator of each fraction by the missing factors. For the first fraction:
\ \[ \frac{w^2 + 3}{(w - 2)(w^2 + 2w + 4)} \times \frac{(w + 2)}{(w + 2)} = \frac{(w^2 + 3)(w + 2)}{(w - 2)(w^2 + 2w + 4)(w + 2)} \ \]
For the second fraction:
\ \[ \frac{2w}{(w - 2)(w + 2)} \times \frac{(w^2 + 2w + 4)}{(w^2 + 2w + 4)} = \frac{2w(w^2 + 2w + 4)}{(w - 2)(w^2 + 2w + 4)(w + 2)} \ \]
Now both fractions have the same denominator, making it easier to combine them.

Expanding and Simplifying

Expanding and simplifying polynomials involves multiplying out expressions and combining like terms. This process can help in breaking down complex expressions and finding their simplest form.
In our example, after rewriting both fractions with a common denominator, we need to expand the numerators and then subtract them:

• Expand \((w^2 + 3)(w + 2) = w^3 + 2w^2 + 3w + 6\)
• Expand \(2w(w^2 + 2w + 4) = 2w^3 + 4w^2 + 8w\)

Then, subtract the numerators: \ \[ w^3 + 2w^2 + 3w + 6 - 2w^3 - 4w^2 - 8w = -w^3 - 2w^2 - 5w + 6 \ \]
Finally, the combined fraction is:
\ \[ \frac{-w^3 - 2w^2 - 5w + 6}{(w - 2)(w^2 + 2w + 4)(w + 2)} \ \]
Make sure to always expand carefully and combine like terms accurately to simplify expressions fully.