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Topic: Arithmetic with Polynomials and Rational Expressions 

Common Core Mathematics 9-12

 Cluster  Perform arithmetic operations on polynomials
  Grade 9-12
   A.APR.1.Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
 Cluster  Understand the relationship between zeros and factors of polynomials
  Grade 9-12
   A.APR.2.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.3.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.
 Cluster  Use polynomial identities to solve problems
  Grade 9-12
   A.APR.4.Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
   A.APR.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. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument]
 Cluster  Rewrite rational expressions
  Grade 9-12
   A.APR.6.Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
   A.APR.7.(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.