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...
Advanced Topics

Complex Numbers and Their Operations

Beyond the Real Numbers

Sometimes, equations like \( x^2 + 1 = 0 \) have no real solutions. Enter complex numbers! A complex number has a real part and an imaginary part, like \( 2 + 3i \), where \( i = \sqrt{-1} \).

Arithmetic with Complex Numbers

  • Addition/Subtraction: Combine real parts and imaginary parts separately.
  • Multiplication: Use distributive property, remembering that \( i^2 = -1 \).
  • Conjugates: The conjugate of \( a + bi \) is \( a - bi \). Useful for dividing.

Visualizing Complex Numbers

They can be plotted on a plane, with the real part on the x-axis and the imaginary part on the y-axis.

Where Are They Used?

Complex numbers pop up in engineering, electronics, and even art!

Key Formula

\[z = a + bi\]

Examples

  • Adding \( (2 + 3i) + (4 + 2i) = 6 + 5i \).

  • Multiplying \( (1 + 2i)(3 - i) = 3 + 6i - i - 2i^2 = 3 + 5i + 2 = 5 + 5i \).

In a Nutshell

Complex numbers extend the real numbers and make it possible to solve equations like \( x^2 + 1 = 0 \).

Key Terms

Imaginary Unit
The symbol \( i \), defined as \( \sqrt{-1} \).
Conjugate
For \( a + bi \), the conjugate is \( a - bi \).
Next
Polynomial Division and the Remainder Theorem