A.APR.2 Know and apply the Ability to make connections Students should be able to identify integer roots of a polynomial Remainder Theorem: For a between factors, roots and equation by graphing. Using these integer roots a student should polynomial p(x) and a number a, evaluating functions then use division to factor until reaching quadratic form allowing the remainder on division by Ability to use both long division them to identify all roots of the polynomial equation. x a is p(a), so p(a) = 0 if and and synthetic division – only if (x a) is a factor of p(x). – (major) A.APR.3 Identify zeros of Students should be able to factor higher degree polynomials that Knowledge of the differences polynomials when suitable in the end behavior of the have rational roots and are therefore easier to factor. factorizations are available, and graphs as dictated by the Students should be able to produce a rough graph of a polynomial use the zeros to construct a leading coefficient and whether that is in factored form. rough graph of the function the function is even or odd defined by the polynomial. Ability to capture the graphical (major)(cross-cutting) behavior of polynomial functions which have roots with multiplicity greater than one Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. A.APR.4 Prove polynomial Knowledge of the process for To prove an identity, logical steps should be used to manipulate identities and use them to proving identities one side of the equation to match the other side of the equation. describe numerical relationships. Ability to see, use and When proving polynomial identities one may not use properties of For example, the polynomial manipulate the structure in equality and must restrict their work to using techniques such as identity an expression as needed to factoring, combining like terms, multiplying polynomials etc. (x2 + y2)2 = (x2 y2)2 + (2xy)2 – prove an identity can be used to generate Pythagorean triples. (additional)
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). (major) Ability to make connections between factors, roots and evaluating functions Ability to use both long division and synthetic division Students should be able to identify integer roots of a polynomial equation by graphing. Using these integer roots a student should then use division to factor until reaching quadratic form allowing them to identify all roots of the polynomial equation. 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. (major)(cross-cutting) Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one Students should be able to factor higher degree polynomials that have rational roots and are therefore easier to factor. Students should be able to produce a rough graph of a polynomial that is in factored form. Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. 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. (additional) Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an expression as needed to prove an identity To prove an identity, logical steps should be used to manipulate one side of the equation to match the other side of the equation. When proving polynomial identities one may not use properties of equality and must restrict their work to using techniques such as factoring, combining like terms, multiplying polynomials etc.
A2_UP_U1_Polynomial_Rational_and_Radical_Relationships_Dec_27.docx