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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.
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.
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.
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.
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.