Cluster: Perform arithmetic operations on polynomials. Featured Mathematical Practice:

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1 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. Perform arithmetic operations on polynomials Understand the definition of a polynomial. Understand the concepts of combining like terms. Add, subtract, and multiply polynomials and understand how closure applies under these operations. Understand the definition of a polynomial Identify what like terms are and what they are not Carry out the required operations for combining like terms Use the distributive property (if necessary) to simplify polynomials in order to identify and combine like terms Distributive property Binomial Trinomial Polynomial Degree

2 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). Understand the relationship between zeros and factors of polynomials Understand and apply the Remainder Theorem. Understand that a is a root of a polynomial function if and only if x-a is a factor of the function. Recognize the relationship between a root/zero and its corresponding factor Perform the operation of division on a polynomial by one of its factors Restate the polynomial in terms of the divisor, quotient, and remainder ( ( ) ( ) ( ) ( )) Root/zero Factor Quotient Remainder Divisor

3 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. Understand the relationship between zeros and factors of polynomials Find the zeros of a polynomial when the polynomial is factored. Use the zeros of a function to sketch a graph of the function. Carry out the factorization of the polynomial Define the coordinates of the x-intercept(s) using the zeros of the polynomial Construct a rough graph using the derived zeros/roots and the lead coefficient of the polynomial Factorization Zeros/roots/x-int. Lead coefficient

4 4. 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. Use polynomial identities to solve problems Understand that polynomial identities include but are not limited to the product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial, etc. Prove polynomial identities by showing steps and providing reasons. Illustrate how polynomial identities are used to determine numerical relationships Identify different polynomial identities o product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial Identify a Pythagorean triple Apply polynomial identities to evaluate numerical relationships o ( ) Pythagorean triple Polynomial identities

5 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. 1 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. Use polynomial identities to solve problems For small values of n, use Pascal s Triangle to determine the coefficients of the binomial expansion. Use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive power. Recognize Pascal s triangle Connect Pascal s Triangle to the Binomial Theorem in order to expand (x + y) n Determine the nth term of an expanded binomial Binomial Pascal s Triangle

6 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. Rewrite rational expressions Rewrite rational expressions, ( ) in the form ( ) ( ) by long division or synthetic division. ( ) ( ) Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding. Use the long-hand division model within integers and write the solution in the form ( ) ( ) ( ) Connect the long-hand division model to polynomials divided by binomial Relate synthetic division to the long hand division model Remainder Theorem Zeros/Roots Synthetic Division

7 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. Rewrite rational expressions Simplify rational expressions by adding, subtracting, multiplying, or dividing. Understand that rational expressions are closed under addition, subtraction, multiplication, and division (by a nonzero expression). Identify like terms (factors) Carryout the factoring process Understand common denominators Transform rational expressions into expressions w/common denominators Simplify rational expressions based on common factors and/or combining like terms Common denominators Factor Combining like terms Rational expression