Problem 98 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\[ \frac{-3a^2 + 7a - 17}{(a+2)(a^2-2a+4)} \]

Step 1 - Factorize the Denominators

Begin by factorizing the denominators where possible. Notice that: 1. \ a^3 + 8 can be expressed as \ (a + 2)(a^2 - 2a + 4).Thus, the denominators become:\[\frac{a-3}{(a+2)(a^2-2a+4)} - \frac{2}{a+2} - \frac{a-3}{a^2-2a+4} \]

Step 2 - Find a Common Denominator

The least common denominator (LCD) is \ (a+2)(a^2-2a+4). Rewrite each fraction with this common denominator:\[ \frac{a-3}{(a+2)(a^2-2a+4)} - \frac{2(a^2-2a+4)}{(a+2)(a^2-2a+4)} - \frac{(a-3)(a+2)}{(a+2)(a^2-2a+4)}\]

Step 3 - Combine the Numerators

Since all fractions have a common denominator, combine the numerators:\[ \frac{(a-3) - 2(a^2 - 2a + 4) - (a-3)(a+2)}{(a+2)(a^2-2a+4)}\]

Step 4 - Expand and Simplify the Numerator

Expand and simplify the numerator expressions:\[ (a - 3) - 2(a^2 - 2a + 4) - (a^2 - a3 + 2a - 6) \]This gives: \[ a - 3 - 2a^2 + 4a - 8 - a^2 - a3 + 2a - 6 \]Combine like terms:\[ -3a^2 + 7a - 17 \]

Step 5 - Final Expression

Write the final simplified expression over the common denominator:\[ \frac{-3a^2 + 7a - 17}{(a+2)(a^2-2a+4)} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It involves breaking down a complex polynomial into simpler, multipliable factors. This technique is useful for simplifying expressions, solving equations, and finding roots.

Consider the polynomial expression in the denominator of our original exercise: \(a^3 + 8\). This can be factored as a sum of cubes: \(a^3 + 2^3 = (a + 2)(a^2 - 2a + 4)\). Recognizing these factorization patterns helps us break down the expression easily.

Another polynomial you should be familiar with is the difference of squares: \(x^2 - y^2 = (x + y)(x - y)\). Factoring converts complex polynomial expressions into simpler, manageable pieces, which is crucial for finding the least common denominator (LCD) and simplifying algebraic fractions.

Least Common Denominator

The least common denominator (LCD) is the smallest expression that can serve as a common denominator for a set of fractions. To find the LCD, we identify the denominators' factors and take the highest power of each factor that appears.

In our example, the factors are \(a + 2\) and \(a^2 - 2a + 4\). The LCD combines these factors: \((a + 2)(a^2 - 2a + 4)\).

With the LCD determined, each fraction in our expression is rewritten to have this common denominator. This step allows us to combine the fractions because they share a common base, simplifying our calculations. Writing fractions in this way ensures accuracy in adding or subtracting algebraic fractions.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their lowest terms. This is done by factoring both the numerator and the denominator, then canceling out common factors.

In our problem, the combined numerator after rewriting with the LCD is: \( (a - 3) - 2(a^2 - 2a + 4) - (a-3)(a+2)\). By expanding and combining like terms, we get: \ -3a^2 + 7a - 17 \.

Our final simplified expression is then written: \ \frac{-3a^2 + 7a - 17}{(a + 2)(a^2 - 2a + 4)} \. Simplifying rational expressions makes complex algebraic operations manageable and clearer.

Always check for further simplifications or factor cancellations to ensure you've reduced the expression to its simplest form.