Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x²+1) + 2x/(x²+1)².

Verified step by step guidance

1. Identify the two rational expressions to be added: $\frac{5x-3}{x^2}$ and $\frac{2x}{{x^2+1}^2}$.
2. Determine the least common denominator (LCD) for the two fractions. Since the denominators are $x^2+1$ and $(x^2+1)^2$, the LCD is $(x^2+1)^2$.
3. Rewrite the first fraction so that it has the LCD as its denominator by multiplying both its numerator and denominator by $(x^2+1)$. This gives: $\frac{(5x-3)(x^2+1)}{{x^2+1}^2}$.
4. Now that both fractions have the same denominator, combine the numerators over the common denominator: $\frac{(5x-3)(x^2+1)\; +\; 2x}{{x^2+1}^2}$.
5. Expand the numerator by distributing and then combine like terms if possible. The expression is now ready for any further simplification or factoring.

Key Concepts

Adding Rational Expressions

Adding rational expressions involves combining fractions with algebraic expressions in the numerator and denominator. When denominators are the same or related, you can add the numerators directly or after adjusting to a common denominator.

Polynomial Denominators and Factoring

Understanding the structure of polynomial denominators, such as recognizing powers like (x^2 + 1) and (x^2 + 1)^2, is essential. This helps in identifying common denominators and simplifying expressions before addition.

Simplifying Algebraic Fractions

After adding rational expressions, simplifying the resulting fraction by combining like terms and factoring if possible is important. This ensures the final answer is in its simplest form for clarity and correctness.