Alaska flagAlaska: Arithmetic with Polynomials and Rational Expressions Math Standards

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ADD, SUBTRACT, AND MULTIPLY POLYNOMIALS. UNDERSTAND THAT POLYNOMIALS FORM A SYSTEM SIMILAR TO THE INTEGERS, NAMELY, THEY ARE CLOSED UNDER THE OPERATIONS OF ADDITION, SUBTRACTION, AND MULTIPLICATION.

    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).

      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.

        PROVE 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.

          (+) 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.

            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.

              (+) ADD, SUBTRACT, MULTIPLY, AND DIVIDE RATIONAL EXPRESSIONS. UNDERSTAND THAT RATIONAL EXPRESSIONS FORM A SYSTEM SIMILAR TO THE RATIONAL NUMBERS, CLOSED UNDER ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION BY A NONZERO RATIONAL EXPRESSION.