Problem 28 - Algebra for College Students

Question

Perform the indicated operations, and express your answers in simplest form. $$ x+\frac{5}{x^{2}-25}-\frac{x^{2}}{x+5} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x + \frac{-x^3 + 5x^2 + 5}{(x-5)(x+5)} \).

Step 1: Identify Common Denominator

The given expression is \( x + \frac{5}{x^2 - 25} - \frac{x^2}{x+5} \). First, factor the denominator of the first fractional term: \( x^2 - 25 = (x-5)(x+5) \). Thus, the common denominator for the fractions is \((x-5)(x+5)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each term with the common denominator of \((x-5)(x+5)\). We'll need to adjust the second fraction:- The first fraction is already \( \frac{5}{(x-5)(x+5)} \).- Rewrite the second fraction: \( \frac{x^2}{x+5} = \frac{x^2(x-5)}{(x+5)(x-5)} \).

Step 3: Combine the Fractions

Combine the fractions over the common denominator:\[\frac{5 - x^2(x-5)}{(x-5)(x+5)}. \]

Step 4: Simplify the Numerator

Distribute \( x^2 \) in the numerator:- \( x^2(x-5) = x^3 - 5x^2 \)- The expression becomes: \( 5 - x^3 + 5x^2 \).Simplify: \(-x^3 + 5x^2 + 5.\)

Step 5: Write the Complete Expression

Now, write down the final simplified expression including the \( x \) term:Combine the whole expression:\[ x + \frac{-x^3 + 5x^2 + 5}{(x-5)(x+5)}. \]

Step 6: Simplify Further if Possible

If it's possible, simplify further any common factors in the numerator and the denominator. In this case, no further simplification can be applied to the expression. So, the simplest form of the expression is:\[ x + \frac{-x^3 + 5x^2 + 5}{(x-5)(x+5)}. \]

Key Concepts

Polynomial Operations

Polynomial operations involve basic arithmetic actions such as addition, subtraction, multiplication, and division. When working with polynomials, finding a common denominator is essential when adding or subtracting fractions. This helps unify the expression so it can be combined into a single fraction.

• To find a common denominator, factor the polynomial denominators fully. For example, here, the expression contains a term with denominator \(x^2 - 25\), which can be factored into \((x-5)(x+5)\).
• Once denominators are uniform, fractions can be combined.

Polynomial operations require translating multi-step processes like representing complex expressions with a single fraction. Quick checks for the need for polynomial simplification can prevent errors in your calculations.

Factoring Quadratics

Factoring quadratics is a crucial skill in simplifying expressions. It involves rewriting a quadratic expression as a product of two binomials. This is particularly helpful when simplifying or combining fractions.

• Consider differences of squares, such as \(x^2 - 25\); this can be rewritten as \((x-5)(x+5)\).
• Recognizing patterns in quadratics can vastly simplify complex algebraic expressions.
• Factoring is a reversible process. This means you can always expand it back to check correctness.

Mastering factoring quadratics provides a solid foundation, enabling more comprehensive operations with polynomials. This understanding aids in tasks like finding common denominators for polynomial fractions.

Simplifying Fractions

Simplifying fractions in algebra often involves polynomial expressions, especially when factoring comes into play. It's about making an expression easier to evaluate or understand.

• The first step is to ensure all expressions have been fully factored. This allows the identification of any potential common factors.
• Combine the fractions using the common denominator found previously.
• Simplify the resulting expression by distributing terms within the numerator, as was done here: \(-x^3 + 5x^2 + 5\).

Simplifying fractions means expressing them in their lowest terms. This makes working with the fractions simpler. Simplification is complete when there is no further reduction possible by canceling common factors from the numerator and the denominator.