Notes on Factoring Polynomials and the Fundamental Theorem of Algebra

Learning Intention
  • Understanding how to factor polynomials and determine when a polynomial is factored completely.
  • Utilizing the Fundamental Theorem of Algebra (FTA) to ascertain the number of solutions for polynomial functions.
Key Concepts
Rational Root Theorem
  • This theorem provides a list of potential solutions for a polynomial function. To find actual solutions, test the values from this list.
  • Example: To solve the polynomial equation x3−8x2+11x+20=0x^3 - 8x^2 + 11x + 20 = 0, utilize the Rational Root Theorem to identify potential rational roots.
Fundamental Theorem of Algebra
  • States that a polynomial of degree nn has exactly nn solutions, which includes real and imaginary solutions.
    • Total Solutions: Number of roots (real + imaginary)
    • Real Solutions: Number of x-intercepts on the graph
    • Imaginary Solutions: Calculated as extIMAG=extTOTAL−extREALext{IMAG} = ext{TOTAL} - ext{REAL}
  • Example: For the polynomial function f(x)=x3+3x2+16x+48f(x) = x^3 + 3x^2 + 16x + 48, identify the number of total, real, and imaginary zeros.
Factoring Techniques
  • Utilize various methods for factoring, including:
    • Greatest Common Factor (GCF)
    • Grouping
    • Synthetic Division
    • Quadratic factoring techniques
Writing Polynomial Functions
  • Polynomial functions can be expressed in different forms:
    • Factored form: f(x)=(x±a)(x+b)f(x)=(x±a)(x+b)
    • Standard form: f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d where the leading coefficient is 1 and coefficients are rational.
  • Example: Writing a polynomial function given zeros such as $10, - ext{√}5$.
Identifying Solutions and Zeros
  • Determine the number of total, real, and imaginary solutions of polynomial functions.
  • Identify real and imaginary zeros from polynomial equations.
  • Example: Given zeros $1, -2, 4i$, construct the corresponding polynomial function.
Solutions of a Polynomial Function
  • The study of polynomial functions encompasses determining their zeros and the behavior at those points.
Repeated Solutions
  • When a factor appears multiple times, it corresponds to repeated solutions.
    1. If the factor x−kx-k is raised to an even degree, the function bounces off the x-axis at that x-intercept.
    2. If the factor x−kx-k is raised to an odd degree, the function crosses the x-axis.
Finding Zeros and Graphing Functions
  • Identify and find zeros of polynomial functions.
  • Examples:
    • For f(x)=2x3+8x2+6xf(x) = 2x^3 + 8x^2 + 6x, find the zeros and sketch the graph.
    • For f(x)=2x3−x2−2x+1f(x) = 2x^3 - x^2 - 2x + 1, find the zeros and sketch the graph.
Formative Assessment
  • Worksheets related to sections 4.5 and 4.6, focusing on solving polynomials by factoring and applying the FTA.
Standards and Goals
  • Align with standards A.APR.B.3 and N.CN.C.9, emphasizing understanding and application of polynomials and the FTA, particularly in real-world contexts such as modeling time on a roller coaster above or below ground.