Algebra

AIII.A.10Prove 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.AIII.A.11Verify the Binomial Theorem by mathematical induction or by a combinatorial argument.AIII.A.12Know 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.AIII.A.13Write rational expressions in simplest form. (For example (x^3 – x^2 – x + 1)/(x^3 + x^2 – x – 1) = (x – 1)/(x + 1).AIII.A.14Decompose a rational function into partial fractions.AIII.A.15Determine asymptotes and holes of rational functions, explain how each was found, and relate these behaviors to continuity.AIII.A.16Add, subtract, multiply and divide rational expressions.AIII.A.17Solve polynomial and rational inequalities. Relate results to the behavior of the graphs.AIII.A.18Find the composite of two given functions and find the inverse of a given function. Extend this concept to discuss the identity function f(x) = x.AIII.A.19Simplify complex algebraic fractions (with/without variable expressions and integer exponents) to include expressing (f(x+h)-f(x))/h as a single simplified fraction when f(x) = 1/1-x for example.AIII.A.20Find the possible rational roots using the Rational Root Theorem.AIII.A.21Find the zeros of polynomial functions by synthetic division and the Factor Theorem.AIII.A.22Graph and solve quadratic inequalities.AIII.A.8Determine characteristics of graphs of parent functions (domain/range, increasing/decreasing intervals, intercepts, symmetry, end behavior, and asymptotic behavior).AIII.A.9Determine the end behavior of polynomial functions.
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