Polynomials and rational expressions

These notes covers polynomials and rational expressions, including their definitions, operations, and applications. Polynomials A polynomial is an algebraic expression composed of terms with variables raised to whole-number exponents. The degree of a polynomial is determined by the highest exponent. Polynomials can be classified as monomials, binomials, or trinomials. • Operations on Polynomials include addition, subtraction, multiplication, and division (using long or synthetic division). • Factoring Polynomials is an essential skill used to simplify expressions and solve equations. Common methods include factoring out the greatest common factor (GCF), difference of squares, trinomial factoring, and sum/difference of cubes. Rational Expressions A rational expression is a fraction in which the numerator and denominator are both polynomials. These expressions can be simplified by factoring and canceling common terms. • Operations on Rational Expressions include addition, subtraction (requiring a common denominator), multiplication, and division (by multiplying by the reciprocal). • Solving Rational Equations involves eliminating denominators by multiplying by the least common denominator (LCD), solving the resulting equation, and checking for extraneous solutions (values that make the denominator zero). Understanding polynomials and rational expressions is crucial for advanced algebra, calculus, and real-world applications such as physics and engineering.