Exponents, Polynomials, and Polynomial Functions

Properties of Exponents

Understanding the properties of exponents is essential for simplifying and manipulating polynomial expressions. These properties allow us to rewrite expressions in simpler or more useful forms.

• Product of Powers Property: When multiplying like bases, add the exponents.

• Quotient of Powers Property: When dividing like bases, subtract the exponents. , where

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

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

• Power of a Quotient Property: Distribute the exponent to both numerator and denominator. , where

• Zero Exponent Property: Any nonzero base raised to the zero power is 1. , where

• Negative Exponent Property: A negative exponent indicates a reciprocal. , where

Example: Simplify

Polynomial Expressions

A polynomial is an algebraic expression consisting of terms in the form , where the exponents are non-negative integers and the coefficients are real numbers.

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

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term without a variable (degree 0).

Example: is a polynomial of degree 3, leading coefficient 4, constant term -5.

Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms (terms with the same variable and exponent).

• Step 1: Arrange terms in standard form (descending order of exponents).

• Step 2: Combine coefficients of like terms.

Example:

Multiplying Polynomials

Multiplying polynomials involves using the distributive property (also known as the FOIL method for binomials) to multiply each term in one polynomial by each term in the other.

• Distributive Property:

• FOIL Method (for binomials): Multiply First, Outer, Inner, Last terms.

Example:

Evaluating Polynomial Functions

To evaluate a polynomial function, substitute the given value for the variable and simplify.

• Step 1: Substitute the value into the polynomial.

• Step 2: Simplify using the order of operations.

Example: Evaluate at

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors. Several methods are used depending on the structure of the polynomial.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms. Example:

• Factoring by Grouping: Group terms to factor common factors, then factor again. Example:

• Trial and Error (Guess and Check): Find factors that multiply to the constant term and add to the middle coefficient (for quadratics).

• AC Method: For , multiply , find factors that sum to , split the middle term, and factor by grouping.

• Difference of Squares:

• Substitution: Substitute a variable to simplify factoring, then back-substitute.

Solving Quadratic Equations

Quadratic equations are equations of the form . There are several methods to solve them:

• Factoring: Set the equation to zero, factor, and set each factor to zero.

• Quadratic Formula:

• Completing the Square: Rewrite the equation in the form and solve for .

• Square Root Property: If , then

Example: Solve by factoring. or or

Summary Table: Factoring Methods

Additional info: This summary covers the main learning objectives for Chapter 5, including properties of exponents, operations with polynomials, factoring techniques, and solving quadratic equations, as outlined in the provided material.