Problem 40 - Algebra for College Students

Question

Simplify each complex fraction. $$\frac{\frac{2}{a^{3}+8}-\frac{3}{a^{2}-2 a+4}}{\frac{4}{a^{2}-4}+\frac{a-3}{a^{3}+8}}$$

Step-by-Step Solution

Answer

Simplified form \[ = \frac{-3a - 4}{5a^2 - 13a + 22} \]

Step 1: Identify and Simplify the Denominators

Factor the denominators where possible. For the denominator terms: For \(a^3 + 8\), it factors as \( (a+2)(a^2 - 2a + 4) \)... For \(a^2 - 4\), it factors as \( (a-2)(a+2) \). So the original fraction becomes: \[ \frac{\frac{2}{(a+2)(a^2-2a+4)}-\frac{3}{a^2-2a+4}}{\frac{4}{(a-2)(a+2)} + \frac{a-3}{(a+2)(a^2-2a+4)}} \]

Step 2: Combine the Numerator and Denominator Terms

To combine fractions in the numerator and the denominator, get a common denominator for each part: For the numerator: The common denominator is \( (a+2)(a^2 - 2a + 4) \). For the denominator: The common denominator is \( (a-2)(a+2)(a^2 - 2a + 4) \).

Step 3: Simplify the Fractions

Combine the fractions in the numerator and denominator: For the numerator: \[ \frac{2 - 3(a+2)}{(a+2)(a^2 - 2a + 4)}; \] Distribute and combine like terms. For the denominator: \[ \frac{4(a^2-2a+4) + (a-3)(a-2)}{(a-2)(a+2)(a^2 - 2a + 4)}; \] Distribute and combine like terms.

Step 4: Final Simplification

Combine the results to get: Numerator: \[ \frac{-3a+2-6}{(a+2)(a^2 - 2a + 4)} = \frac{-3a-4}{(a+2)(a^2 - 2a + 4)} \] Denominator: \[ \frac{4a^2-8a + 16 + a^2 -5a +6} {(a-2)(a+2)(a^2 - 2a + 4)} = \frac{5a^2-13a + 22} {(a-2)(a+2)(a^2 - 2a + 4)} \] Finally divide the numerator by the denominator: \[ \frac{(-3a - 4)/(a+2)(a^2-2a+4)} * \frac{(a-2)(a+2)(a^2 -2a + 4)}{5a^2 - 13a + 22} = \frac{-3a - 4}{5a^2 - 13a + 22} \]

Key Concepts

factoring polynomials

When working with complex fractions, one of the first steps to simplifying them is to factor the polynomials in the denominators. Factoring makes it easier to see common terms and cancel them out.

For instance, in the given exercise:

• Factor \(a^3 + 8\): Notice that \(a^3 + 8\) can be factored as a sum of cubes, which gives us \( (a+2)(a^2 - 2a + 4) \).



• Factor \(a^2 - 4\): This is a difference of squares and can be factored into \( (a-2)(a+2) \).

By factoring these, the complex fraction becomes simpler, making it easier to find common denominators and combine terms.

common denominator

A crucial step in simplifying complex fractions is finding a common denominator for the fractions in both the numerator and the denominator. This helps in combining the fractions effectively.

For example, in our problem:

• For the numerator fractions \( \frac{2}{(a+2)(a^2-2a+4)} \) and \( \frac{3}{a^2-2a+4} \): The common denominator here would be \( (a+2)(a^2 - 2a + 4) \).



• For the denominator fractions \( \frac{4}{(a-2)(a+2)} \) and \( \frac{a-3}{(a+2)(a^2-2a+4)} \): The common denominator is \( (a-2)(a+2)(a^2 - 2a + 4) \).

This allows us to rewrite each fraction with a common denominator, making it easier to combine them later.

combining fractions

After finding common denominators, the next step is to combine the fractions in both the numerator and the denominator of the complex fraction.

This process involves:

• For the numerator: Combine \( \frac{2}{(a+2)(a^2 - 2a + 4)} \) and \( \frac{3}{a^2 - 2a + 4} \) into a single fraction:
  \[ \frac{2 - 3(a+2)}{(a+2)(a^2 - 2a + 4)} \].



• For the denominator: Combine \( \frac{4}{(a-2)(a+2)} \) and \( \frac{a-3}{(a+2)(a^2 - 2a + 4)} \) into:
  \[ \frac{4(a^2-2a+4) + (a-3)(a-2)}{(a-2)(a+2)(a^2 - 2a + 4)} \].

Once the like terms are combined, it simplifies the fractions further and makes it easier to proceed with the final simplification steps.

simplification steps

With the fractions combined and simplified, the final step involves additional simplification:

• Simplify the numerator: Combine and simplify the terms in \( \frac{2 - 3(a+2)}{(a+2)(a^2 - 2a + 4)} \) to get \( \frac{-3a - 4}{(a+2)(a^2 - 2a + 4)} \).



• Simplify the denominator: Combine and simplify the terms in \(\frac{4(a^2-2a+4) + (a-3)(a-2)}{(a-2)(a+2)(a^2 - 2a + 4)} \) to get \( \frac{5a^2-13a + 22} {(a-2)(a+2)(a^2 - 2a + 4)} \).

The last step is to divide the simplified numerator by the simplified denominator. This results in:
\[ \frac{(-3a-4)/(a+2)(a^2-2a+4)} \div \frac{5a^2-13a+22}{(a-2)(a+2)(a^2-2a+4)} \] Combining the fractions, this simplifies to:
\[ \frac{-3a - 4}{5a^2 - 13a + 22} \].

This final fraction is the simplified form of the original complex fraction.