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