Topic: Arithmetic with Polynomials and Rational Expressions |
Common Core Mathematics 9-12 |
| Cluster Perform arithmetic operations on polynomials |
| | Grade 9-12 |
| | | A.APR.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. |
| Cluster Understand the relationship between zeros and factors of polynomials |
| | Grade 9-12 |
| | | 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). |
| | | 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. |
| Cluster Use polynomial identities to solve problems |
| | Grade 9-12 |
| | | A.APR.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. |
| | | A.APR.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] |
| Cluster Rewrite rational expressions |
| | Grade 9-12 |
| | | A.APR.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. |
| | | A.APR.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. |