Big Idea:

Big Idea 3

The structure of polynomial graphs and equations gives insights into their roots.

1 week

Evidence of Understanding

  • analyze connections between the zeros of the polynomial function and its graph
    • approximate the zeros of a polynomial function from its graph and verify them algebraically
    • discover and describe patterns for even or odd multiplicity of zeros
    • create a polynomial function (in factored form) from the zeros on its graph
    • verify a zero by algebraically showing that the product of the factors are equivalent to the polynomial’s function rule (in cases where the lead coefficient is one)
      • for zeros j, k, and p,  show that (x - j)(x - k)(x - p) = x3 + bx2 + cx + d
      • add, subtract, and multiply complex numbers
         
  • generalize characteristics about the zeros of any polynomial function and use them to solve problems
    • make and evaluate predictions for possible roots of a polynomial from its equation
    • describe patterns and properties relating the quantities in a quadratic, cubic, or quartic equation and its roots
      • Example: students can discover, show, and explain that if 1 is a root for any quadratic, ax2 + bx + c, then the sum of a, b, and c must be zero
      • possible extension: discover and use the Rational Zeros Theorem to identify all the possible rational zeros of a function
    • explain why the Remainder Theorem can prove whether any value is a zero for the function
      • use long division to verify the roots of a polynomial with the Remainder Theorem
      • possible extension: understand how synthetic division is related to long division and can use it to also test and determine roots for any polynomial equation
    • compare the properties of two functions/equations according to their zeros/roots (or vice versa)
    • use characteristics of a polynomial equation to determine and justify methods for finding roots
      • solve for and state all the roots and corresponding factors using a variety of methods including analysis of a graph or table in addition to long division, factoring, etc.  

Develop conceptual understanding:

multiplicity, factor, long division, Remainder Theorem, (+)Synthetic Division, (+)Fundamental Theorem of Algebra

Supporting terms to communicate:

roots, zeros, coefficient, perfect squares, difference of squares, imaginary, complex 

Core Resource
A core resource supports multiple days of instruction.