Rational and Polynomial Functions

Use the structure of an expression to identify ways to rewrite it. Tasks are limited to polynomial, rational, or exponential expressions.  For example, see x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).   

Write expressions in equivalent forms to solve problems.  Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Understand the relationship between zeros and factors of polynomials.  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.​  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. Tasks include quadratic, cubic, and quartic polynomials and polynomials in which factors are not provided.  For example, find the zeros of f(x) = (x² - 1)(x² + 1).

Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Understand solving equations as a process of reasoning and explain the reasoning.  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Represent and solve equations and inequalities graphically.  Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Rewrite rational expressions.  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.  From the PARCC Model Content Frameworks: “This standard sets an expectation that students will divide polynomials with remainder by inspection in simple cases. For example, one can view the rational expression ^(x+4)/_(x+3) as ^(x+4)/_(x+3) = ^(x+3)+1/_(x+3) = 1 + ¹/_(x+3) ^."

Analyze functions using different representations. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Perform arithmetic operations with complex numbers.  Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

Perform arithmetic operations with complex numbers.  Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions.

Use polynomial identities to solve problems.  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.

Solve Systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Solve Systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.

Translate between the geometric description and the equation for a conic section. Derive the equation of a parabola given a focus and directrix.

Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.