Arithmetic with Polynomials and Rational Expressions
|Perform arithmetic operations on polynomials. [Beyond quadratic]|
|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.
|Understand the relationship between zeros and factors of polynomials.|
|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).|
|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.
|Use polynomial identities to solve problems.|
|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.
|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.
|Rewrite rational expressions. [Linear and quadratic denominators]|
|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.
|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.