HSA-APR.C.4
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- HSA-APR.C.4Prove 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 Pascalâ€™s Triangle.
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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.
29 Lesson(s)
Proving Polynomial Identities
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(7)
Resources
(18)
Reflections
(1)
Review of Polynomial Roots and Complex Numbers
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(2)
Resources
(10)
The Remainder Theorem
Algebra II »
Unit: Polynomial Theorems and Graphs
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).
Standards:
Favorites
(5)
Resources
(13)
Reflections
(1)
Quiz and Intro to Graphs of Polynomials
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(3)
Resources
(12)
Reflections
(1)
Seeing Structure in Expressions - Factoring Higher Order Polynomials
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(6)
Resources
(25)
Reflections
(2)
Polynomial Long Division and Solving Polynomial Equations
Algebra II »
Unit: Polynomial Theorems and Graphs
Algebra II »
Unit: Polynomial Theorems and Graphs
Big Idea:
Operations with polynomials are a lot like operations with integers.
Standards:
Favorites
(7)
Resources
(22)
Reflections
(1)
The Fundamental Theorem of Algebra and Imaginary Solutions
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Resources
(14)
Reflections
(1)
Arithmetic with Complex Numbers
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(5)
Resources
(23)
Reflections
(2)
Unit Test: Sequences and Series
Algebra II »
Unit: Sequences and Series
Algebra II »
Unit: Sequences and Series
Performance Task - Representing Polynomials
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(2)
Resources
(15)
Reflections
(1)
Graphing Polynomials - End Behavior
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(8)
Resources
(15)
Reflections
(1)
Big Idea:
Show your students that they really can "prove it" with this lesson on polynomial identities.
Standards:
Favorites
(1)
Resources
(10)
Reflections
(1)
Big Idea:
First you create it, then you use it...this lesson allows students to use the polynomial identities they just proved.
Standards:
Resources
(14)
Reflections
(1)
Graphing Polynomials - Roots and the Fundamental Theorem of Algebra
Algebra II »
Unit: Polynomial Theorems and Graphs
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.
Standards:
Favorites
(6)
Resources
(20)
Reflections
(1)
Big Idea:
Egyptian fractions provide an interesting arena for putting rational functions to use.
Standards:
Favorites
(1)
Resources
(9)
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