Problem 94 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{8 a}{2 a^{2}+4 a+2}-\frac{3 a-3}{a^{2}-1} $$

Step-by-Step Solution

Answer

\(\frac{a}{(a+1)^2}\)

Step 1: Factorize the Denominators

First, factorize the denominators of both fractions. For the first fraction: The denominator is: \[ 2a^2 + 4a + 2 \] Factor out the greatest common factor (2): \[ 2(a^2 + 2a + 1) = 2(a + 1)^2 \] For the second fraction: The denominator is: \[ a^2 - 1 \] This is a difference of squares: \[ (a+1)(a-1) \]

Step 2: Rewrite the Fractions

Rewrite the fractions with the factored forms of their denominators: \[ \frac{8a}{2(a+1)^2} - \frac{3a-3}{(a+1)(a-1)} \]

Step 3: Find the Common Denominator

The common denominator is the least common multiple of \[ 2(a+1)^2 \] and \[ (a+1)(a-1) \]. The common denominator is: \[ 2(a+1)^2(a-1) \]

Step 4: Adjust the Numerators

Rewrite each fraction to have the common denominator: The first fraction: \[ \frac{8a}{2(a+1)^2} = \frac{8a(a-1)}{2(a+1)^2(a-1)} = \frac{8a(a-1)}{2(a+1)^2(a-1)} \] The second fraction: \[ \frac{3a-3}{(a+1)(a-1)} = \frac{2(a+1)(3a-3)}{2(a+1)^2(a-1)} \]

Step 5: Combine the Fractions

Combine the fractions over the common denominator: \[ \frac{8a(a-1) - 2(a+1)(3a-3)}{2(a+1)^2(a-1)} \]

Step 6: Simplify the Numerator

Expand and simplify the numerator: First, expand: \[ 8a(a-1) - 2(a+1)(3a-3) = 8a^2 - 8a - 2(3a^2 - 3a + 3a - 3) \] Combine like terms: \[ 8a^2 - 8a - 6a^2 + 6a = 2a^2 - 2a \] So the simplified numerator is: \[ 2a(a-1)\]

Step 7: Simplify the Fraction

The final fraction after simplification: \[ \frac{2a(a-1)}{2(a+1)^2(a-1)} \] Cancel out common factors: \[ \frac{a}{(a+1)^2} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. It involves breaking down a polynomial into products of smaller polynomials.

For example, in our exercise, we factorize the denominators:

• The first denominator was: \(2a^2 + 4a + 2\)
• We factored out the greatest common factor (GCF), which is 2: \[ 2(a^2 + 2a + 1) \]
• Then further simplified it to: \[ 2(a + 1)^2 \]

Similarly, for the second denominator:

• The given polynomial was: \(a^2 - 1\)
• This is recognized as a difference of squares and can be factored to: \( (a + 1)(a - 1) \)

By factoring polynomials, we make it easier to find common denominators and simplify the entire fraction.

Finding Common Denominators

Finding a common denominator is essential when adding or subtracting rational expressions. The common denominator should be the least common multiple (LCM) of the denominators.

In our exercise, we have:

• The factored form of the first denominator is \(2(a + 1)^2\)
• The factored form of the second denominator is \((a + 1)(a - 1)\)

To find the LCM, we look for the highest power of each factor present in the factorizations:

• Here, the common denominator combines: \[ 2(a+1)^2(a-1) \]

With a common denominator in place, we can rewrite each fraction to share this common base, allowing us to subtract the fractions and combine them into a single expression.

Simplifying Fractions

Simplifying fractions is the final step in dealing with rational expressions. The goal is to reduce the fraction to its simplest form.

After combining fractions over the new common denominator in our exercise, we need to simplify the numerator:

• First, expand and simplify the expression: \[ 8a(a-1) - 2(a+1)(3a-3) \]
• After expansion: \[ 8a^2 - 8a - 6a^2 + 6a \]
• Combine like terms to arrive at: \[ 2a(a - 1) \]

The simplified numerator and the common denominator are then combined into one fraction:
\[ \frac{2a(a-1)}{2(a+1)^2(a-1)} \]
Finally, we cancel out common factors:
\[ \frac{a}{(a+1)^2} \]
This process reduces the fraction to its simplest, most efficient form, making it much easier to understand and work with.