A-APR. Arithmetic with Polynomials and Rational Expressions

A-APR.A. Perform arithmetic operations on polynomials.

A-APR.A.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.

• Non-Negative Polynomials
• Powers of 11

A-APR.B. Understand the relationship between zeros and factors of polynomials.

A-APR.B.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 $p(x)$.

• Graphing from Factors III
• The Missing Coefficient
• Zeroes and factorization of a general polynomial
• Zeroes and factorization of a non polynomial function
• Zeroes and factorization of a quadratic polynomial I
• Zeroes and factorization of a quadratic polynomial II

A-APR.B.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.

• Graphing from Factors I
• Graphing from Factors II
• Graphing from Factors III
• Solving a Simple Cubic Equation

A-APR.C. Use polynomial identities to solve problems.

A-APR.C.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.

• Trina's Triangles

A-APR.C.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.*

• Powers of 11

A-APR.D. Rewrite rational expressions.

A-APR.D.6. Rewrite simple rational expressions in different forms; write $\frac{a(x)}{b(x)}$ in the form $q(x) + \frac{r(x)}{b(x)}$, 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.

• Combined Fuel Efficiency
• Egyptian Fractions II

A-APR.D.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.

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