Add or subtract, as indicated. 5/x + 2 + 2/x² - 2x + 4 - 60/x³ + 8

Verified step by step guidance

1. Identify and group like terms based on their variable parts. The terms are: \(\frac{5}{x}\), \$2\(, \)\frac{2}{x^2}\(, \)-2x\(, \)4\(, \)-\frac{60}{x^3}\(, and \)8\(. Group constants together, terms with \)x\( in the denominator together, and terms with \)x$ in the numerator separately.
2. Combine the constant terms: \$2 + 4 + 8$. This will simplify the constant part of the expression.
3. Write the terms with \(x\) in the denominator separately: \(\frac{5}{x}\), \(\frac{2}{x^2}\), and \(-\frac{60}{x^3}\). These cannot be combined directly because they have different powers of \(x\) in the denominator.
4. Keep the term with \(x\) in the numerator, \(-2x\), separate since it does not combine with any other term.
5. Rewrite the expression as the sum of the simplified constants, the separate fractional terms with different powers of \(x\) in the denominator, and the \(-2x\) term. This is the simplified form of the expression.

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 with polynomials in the numerator and denominator. To add or subtract them, you must find a common denominator, rewrite each term with this denominator, and then combine the numerators accordingly.

Polynomial Degree and Terms

Understanding the degree of polynomial terms (the highest power of the variable) helps in organizing and simplifying expressions. Terms like 5/x, 2/x², and 60/x³ involve negative exponents when rewritten, which affects how they are combined with polynomial terms.