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- HSA-APR.C.4Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = (x<sup>2</sup> - y<sup>2</sup>)<sup>2</sup> + (2xy)<sup>2</sup> can be used to generate Pythagorean triples.
- HSA-APR.C.5(+) Know and apply the Binomial Theorem for the expansion of (x + y)<sup>n</sup> in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

Proving Polynomial Identities

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:There are links between polynomials and geometry. Both branches of math use proof and a polynomial identity can be used to generate Pythagorean triples.

Review of Polynomial Roots and Complex Numbers

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:Arithmetic with polynomials and complex numbers are performed in similar ways, but powers of i have a meaning distinct from powers of variables.

The Remainder Theorem

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:Any polynomial p(x) can be written as a product of (x – a) and some quotient q(x), plus the remainder p(a).

Quiz and Intro to Graphs of Polynomials

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:The graph of a polynomial can look a variety of ways depending on the degree, lead coefficient, and linear factors.

Seeing Structure in Expressions - Factoring Higher Order Polynomials

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:Like quadratic expressions, some higher order polynomial expressions can be rewritten in factored form to reveal values that make the expression equal to zero.

Polynomial Long Division and Solving Polynomial Equations

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:Operations with polynomials are a lot like operations with integers.

The Fundamental Theorem of Algebra and Imaginary Solutions

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:The degree of a polynomial equation tells us how many solutions to expect as long as we include both real and imaginary solutions.

Arithmetic with Complex Numbers

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:Imaginary numbers are used to represent quantities that have two parts; working with these numbers is similar to working with polynomial expressions.

Performance Task - Representing Polynomials

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:The graphs of polynomial functions can be transformed by altering the algebraic form in specific ways. These transformations have the same effect on all the function types studied in Algebra 2 and Precalculus.

Graphing Polynomials - End Behavior

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:A good sketch of a polynomial function can be produced by considering the end-behavior, roots and y-intercept of a polynomial function.

Prove It

Algebra II

» Unit:

Algebraic Arithmetic

Big Idea:Show your students that they really can "prove it" with this lesson on polynomial identities.

Use It

Algebra II

» Unit:

Algebraic Arithmetic

Big Idea:First you create it, then you use it...this lesson allows students to use the polynomial identities they just proved.

Graphing Polynomials - Roots and the Fundamental Theorem of Algebra

Algebra II

» Unit:

Polynomial Theorems and Graphs

Big Idea:A good sketch of a polynomial function can be produced by considering the end-behavior, roots and y-intercept of a polynomial function.

Egyptian Fractions

Algebra II

» Unit:

Rational Functions

Big Idea:Egyptian fractions provide an interesting arena for putting rational functions to use.

HSA-APR.C.4

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^{2} + y^{2})^{2} = (x^{2} - y^{2})^{2} + (2xy)^{2} can be used to generate Pythagorean triples.

HSA-APR.C.5

(+) Know and apply the Binomial Theorem for the expansion of (x + y)^{n} in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.