Exponent Rules and Polynomials

Introduction

This lesson covers the fundamental rules for working with exponents and introduces basic operations with polynomials. Mastery of these concepts is essential for success in precalculus, as they form the foundation for more advanced algebraic manipulations and problem-solving.

Rules of Exponents

Exponent rules allow us to simplify expressions involving powers. Understanding these rules is crucial for manipulating algebraic expressions efficiently.

• Product Rule: When multiplying like bases, add the exponents.

• Quotient Rule: When dividing like bases, subtract the exponents.

• Power of a Power Rule: When raising a power to another power, multiply the exponents.

• Power of a Product Rule: Distribute the exponent to each factor inside the parentheses.

• Power of a Quotient Rule: Distribute the exponent to both numerator and denominator.

• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive exponent.

• Zero Exponent Rule: Any nonzero base raised to the zero power is 1. (for )

Examples and Applications

Simplifying Exponential Expressions

To simplify expressions with exponents, apply the rules above systematically. Always write final answers without negative exponents unless specified otherwise.

• Example: Simplify Solution: Multiply coefficients and add exponents for like bases.

• Example: Simplify Solution: Apply the power of a quotient and product rules.

Polynomials: Definitions and Operations

A polynomial is an algebraic expression consisting of terms with variables raised to non-negative integer exponents. Operations with polynomials include addition, subtraction, and multiplication.

• Degree of a Polynomial: The highest exponent of the variable in the polynomial.

• Like Terms: Terms with the same variable(s) raised to the same power(s).

• Combining Like Terms: Add or subtract coefficients of like terms.

• Example: Simplify Solution: Distribute the negative sign and combine like terms.

• Multiplying Polynomials: Use the distributive property (FOIL for binomials) to multiply each term in one polynomial by each term in the other. Example: Solution: Distribute each term and combine like terms. Combine like terms:

Summary Table: Exponent Rules

Additional info: These notes provide foundational skills for algebraic manipulation, which are essential for all subsequent topics in precalculus, including functions, rational expressions, and polynomial equations.