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 2

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

1 week
See Standards
Common Core: Major Standards:
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-REI.B.4b

Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

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.

N-CN.C.7

Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions.

Evidence of Understanding

  • algebraically determine the rational roots of a factorable polynomial (primarily quadratics)
    • notice patterns that make a quadratic equation factorable or un-factorable
      • describe the relationship between a quadratic equation’s roots and factors
      • show a quadratic is factorable when there is no constant remaining and the original form of the equation is evenly divisible by either factor
    • identify perfect squares and difference of squares (including even degree polynomials equations)
      • recognize when factors correspond to imaginary roots
  • describe irrational, imaginary, or complex roots of a quadratic equation
    • explain when a root is imaginary or irrational using a function’s degree and its graph or table
    • approximate the value of an irrational root by analyzing a function’s graph or table
    • calculate irrational, imaginary, or complex roots from an equation in standard or vertex form  
      • simplify a radical and use perfect squares to predict its location on a number line
      • simplify a radical that involves an imaginary factor
    • verify both roots must be imaginary if one root is imaginary (also verify for complex)
  • generalize characteristics about the roots of any quadratic equation and use them to solve problems
    • explain why perfect square trinomials have one unique root
    • justify when completing the square is an ideal strategy for identifying roots
      • create visual models for a given quadratic equation
      • connect parts of a visual model with terms or quantities in its quadratic equation
      • derive the quadratic formula from the process of completing the square
    • describe roots of a quadratic equation using the discriminant
    • use given roots to describe features of the parabola or create a viable quadratic equation
      • add, subtract, or multiply imaginary or complex numbers and explain the process
    • determine all the roots using a variety of methods including analysis of a graph or table in addition to completing the square, factoring, or the quadratic formula

Develop conceptual understanding:

imaginary, complex, completing the square, quadratic formula, discriminant

Supporting terms to communicate:

roots, real, rational, irrational, perfect square, factor, vertex form, standard form, factored form, inverse operations 

Core Resource

A core resource supports multiple days of instruction.