Add or subtract, as indicated. 4/x + 1 + 1/x² - x + 1 - 12/x³ + 1

Verified step by step guidance

1. Identify and group like terms based on their powers of \(x\). The terms are: \(\frac{4}{x}\), \$1\(, \)\frac{1}{x^2}\(, \)-x\(, \)1\(, \)-\frac{12}{x^3}\(, and \)1$.
2. Combine the constant terms: \$1 + 1 + 1$.
3. Write the expression grouping terms by powers of \(x\): \(-\frac{12}{x^3} + \frac{1}{x^2} + \frac{4}{x} - x + (1 + 1 + 1)\).
4. Since the terms have different powers of \(x\), the expression cannot be simplified further by combining unlike terms; just write the simplified constants and keep the rest as is.
5. Present the final simplified expression as a sum of terms with constants combined and other terms in their fractional or polynomial form.

Key Concepts

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This simplifies expressions by reducing the number of terms. For example, terms with x² can be combined, but terms with x and x² cannot.

Operations with Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, denominator, or both. Adding or subtracting them requires a common denominator. Once the denominators match, you combine the numerators accordingly.

Polynomial Arithmetic

Polynomial arithmetic includes adding, subtracting, and multiplying polynomials by combining like terms and applying distributive properties. Understanding how to manipulate polynomials is essential when simplifying expressions involving multiple terms with variables.