Rational and Polynomial Functions

9 Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
A2 U4: Rational and Polynomial Functions
Browse Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 3

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

1 week
See Standards
Common Core: Major Standards:
A-APR.B.2

Understand the relationship between zeros and factors of polynomials.  Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

A-APR.B.3

Understand the relationship between zeros and factors of polynomials.​  Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Tasks include quadratic, cubic, and quartic polynomials and polynomials in which factors are not provided.  For example, find the zeros of f(x) = (x2 - 1)(x2 + 1).

A-REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-SSE.A.2 Algebra II

Use the structure of an expression to identify ways to rewrite it. Tasks are limited to polynomial, rational, or exponential expressions.  For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).   

Common Core: Additional Standards:
N-CN.A.1

Perform arithmetic operations with complex numbers.  Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

N-CN.A.2

Perform arithmetic operations with complex numbers.  Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Common Core: Plus Standards:
N-CN.C.9(+)

Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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.