LAC 28:CXLII.3105. Arithmetic with Polynomials and Rational Expressions

A. Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system comparable to the integers, as they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. B. Understand the relationship between zeros and factors of polynomials. 1. 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). 2. 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. C. Use polynomial identities to solve problems. 1. Describe numerical relationships using polynomial identities. 2. 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. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. D. Rewrite rational expressions. 1. 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 problems, a computer algebra system. 2. Perform operations on rational expressions, building from previous knowledge that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.