Algebra II

Advanced algebraic concepts including polynomials, rational expressions, and complex numbers.

Polynomials and Their PropertiesFactoring and Solving Polynomial EquationsRational Expressions and Equations
Complex Numbers and Their OperationsPolynomial Division and the Remainder TheoremPolynomial and Rational Inequalities
Modeling Real-World Data with PolynomialsApplications of Rational Expressions in Science and EngineeringComplex Numbers in Electrical Engineering and Art
Mastering Word ProblemsChecking Your SolutionsSmart Use of Calculators and Technology
Polynomials and Thei...Factoring and Solvin...Rational Expressions...Complex Numbers and ...Polynomial Division ...Polynomial and Ratio...Modeling Real-World ...Applications of Rati...Complex Numbers in E...Mastering Word Probl...Checking Your Soluti...Smart Use of Calcula...
Basic Concepts

Polynomials and Their Properties

What Are Polynomials?

Polynomials are algebraic expressions that combine numbers and variables using only addition, subtraction, and multiplication, along with non-negative integer exponents. Each piece, like \( 3x^2 \) or \( -5x \), is called a term.

Parts of a Polynomial

  • Degree: The highest exponent of the variable (e.g., \( 4x^3 \) has degree 3).
  • Leading Coefficient: The coefficient of the term with the highest degree.
  • Constant Term: The term without a variable.

Operations with Polynomials

You can add, subtract, and multiply polynomials just like regular numbers, as long as you combine like terms (terms that have the same variables and exponents).

Why Do Polynomials Matter?

Polynomials model everything from the path of a thrown ball to population growth.

Key Formula

\[P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\]

Examples

  • The area of a rectangle with length \( x + 2 \) and width \( x \) is \( x(x + 2) = x^2 + 2x \).

  • The equation \( 2x^2 - 3x + 1 \) represents a quadratic polynomial.

In a Nutshell

Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication.

Key Terms

Term
A single part of a polynomial, like \( 5x \) or \( -3x^2 \).
Degree
The highest exponent in a polynomial.
Coefficient
A number multiplying the variable in a term.
Next
Factoring and Solving Polynomial Equations