Algebra 3/4

Algebra 3/4 delves into advanced algebraic concepts, including polynomial functions, rational expressions, and complex numbers.

Polynomial FunctionsRational ExpressionsComplex Numbers
Polynomial Division and the Remainder TheoremRoots and Zeros of PolynomialsOperations with Complex Numbers
Modeling Real-Life Situations with PolynomialsRational Functions in Science and EngineeringComplex Numbers in Technology
Breaking Down ProblemsChecking Your Answers
Polynomial FunctionsRational ExpressionsComplex NumbersPolynomial Division ...Roots and Zeros of P...Operations with Comp...Modeling Real-Life S...Rational Functions i...Complex Numbers in T...Breaking Down Proble...Checking Your Answer...
Advanced Topics

Roots and Zeros of Polynomials

Finding the Roots

The roots (or zeros) of a polynomial are the values of \( x \) that make the polynomial equal zero. These are where the graph crosses the x-axis.

Methods for Finding Roots

  • Factoring: Write the polynomial as a product of factors and set each to zero.
  • Quadratic Formula: For degree 2 polynomials, use \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Graphing: Sometimes, you can estimate roots by looking at the graph.

Multiplicity

If a root appears more than once, it has multiplicity. This affects how the graph behaves at that root: it may just touch the axis instead of crossing it.

Applications

Finding roots helps solve equations, predict when things will hit zero (like a ball hitting the ground), and more.

Examples

  • The roots of \( x^2 - 5x + 6 \) are 2 and 3.

  • The graph of \( (x - 2)^2 \) touches the x-axis at \( x = 2 \) but doesn't cross.

In a Nutshell

Roots are the solutions to polynomial equations where the function equals zero.

Next
Operations with Complex Numbers