Algebra II

Professor Sellers explains the topics covered in the series; the importance of algebra and how one can get the most out of these episodes; the fundamentals of algebra; reviewing the order of operations and trying a hand at several problems.

Exploring linear equations; studying one-step equations and then advancing to those requiring two or more steps to solve; applying the distributive property to simplify certain problems; learning about the three categories of linear equations.

Taking knowledge of linear equations a step ahead; examples involving absolute values, which can be thought of as a distance on a number line, always expressed as a positive value; recognizing absolute value problems that have limited or no answers.

Moving into the visual realm; how linear equations are represented as straight lines on graphs using either the slope-intercept or point-slope forms of the function; parallel and perpendicular lines; identifying them by the value of their slopes.

Reversing the procedure from the previous episode; starting with an equation and drawing the line that corresponds to it; matching four linear equations to their graphs; how to rewrite an equation to move its graph in different directions.

Functions are important not only for algebra, but for precalculus, calculus and higher mathematics; a function's definition; concepts such as domain and range; the vertical line test for determining whether a given curve is a graph of a function.

Solving systems of two linear equations by graphing the corresponding lines; looking for the intersection point; discovering that there are three possible outcomes; no solution, infinitely many solutions and exactly one solution.

Two other techniques for solving systems of two linear equations: the method of substitution solves one of the equations and substitutes the result into the other; elimination adds or subtracts the equations to see if a variable can be eliminated.

As the number of variables increases, it becomes unwieldy to solve systems of linear equations by graphing; these problems are not as hard as they look; systems of three linear equations often yield to the strategy of eliminating variables.

Making the leap into systems of linear inequalities, where the solution is a set of values on one side or another of a graphed line; an inequality is an assertion such as less than or greater than, which encompasses a range of values.

Begin the investigation of quadratic functions by visualizing what they look like when graphed; they form a U-shaped curve called a parabola, whose location on the coordinate plane can be predicted based on the individual terms of the equation.

One of the most important skills related to quadratics is factoring; reviewing the basics of factoring; learning to recognize a special case known as the difference of two squares; working on a word problem that translates into a quadratic equation.

The square root approach to solving quadratic equations works not just for perfect squares, but also for values that do not seem to involve squares at all; probing the idea behind this technique; venturing into the world of complex numbers.

Completing the square involves turning a quadratic equation into a solvable form that includes a perfect square; a benefit of this approach is that the rewritten form gives the coordinates for the vertex of the parabola represented by the equation.

When other approaches fail, the quadratic formula can solve every quadratic equation; practicing this formula on a wide range of problems; how a special expression called the discriminant tells how many real-number solutions the equation has.

Stepping into the realm of quadratic inequalities, where the boundary graph is not a straight line but a parabola; using the skills for analyzing quadratic expressions to sketch graphs quickly and solve systems of quadratic inequalities.

Delving into the algebra of conic sections, which are cross-sectional shapes produced by slicing a cone; studying parabolas and hyperbolas, which differ in how many variable terms are squared in each; how to sketch a hyperbola from its equation.

The algebraic properties of ellipses and circles; ellipses resemble stretched circles; they are defined by their major and minor axes, whose ratio determines the ellipses' eccentricity; circles are ellipses whose eccentricity equals one.

Pausing to examine the nature of polynomials, a class of algebraic expressions, Professor Sellers introduces several useful concepts, such as the standard form of polynomials and their degree, domain, range and leading coefficients.

Graphing polynomial functions to see how they differ from non-polynomials; how the general shape of the graph can be predicted from the highest exponent of the polynomial, known as its degree; how other terms in the function also affect the graph.

The algebraic side of polynomial functions; learning how to combine them in many different ways, including addition, subtraction, multiplication and even long division; discovering which of these operations do and do not produce new polynomials.

How to solve polynomial equations where the degree is greater than two by turning them into expressions one already knows how to handle; the "toolbox" includes techniques such as the difference of two squares and the difference of two cubes.

Going beyond the approaches learned so far; how to solve polynomial equations by applying two tools for finding rational roots, which are the rational roots theorem and the factor theorem; both will prove useful in succeeding lessons.

Exploring two additional tools for identifying the roots of polynomial equations; Descartes' rule of signs, which narrows down the number of possible real roots; the fundamental theorem of algebra, which gives the total of all roots for a polynomial.

Shifting gears away from polynomials to focus on expressions involving roots, including square roots, cube roots and roots of higher degrees; practicing multiplying, dividing, adding and subtracting a wide variety of radical expressions.

Trying one's hand at solving equations with roots and radicals; beginning by learning how to manipulate rational, or fractional, exponents; practicing with simple equations while being on the lookout for extraneous, or "imposter", solutions.

Experimenting with curves formed by simple radical functions; determining the domain of the function, which tells the general location of the graph on the coordinate plane; how different terms in the function alter the graph in predictable ways.

Studying graphs of rational functions, which are the ratio of two polynomials; these graphs are more complicated; their characteristics can be determined by calculating the domain, the x- and y-intercepts, and the vertical and horizontal asymptotes.

Combining rational functions using addition, subtraction, multiplication, division and composition; the trick is to start each problem by putting the expressions in factored form; leaving the answer in factored form also allows for other operations.

Trying the procedure of splitting a more complicated rational expression into its component parts, called partial fraction decomposition, this approach is a topic in introductory calculus; it is used for solving more advanced math problems.

Exponential functions are important in real-world applications; experimenting with exponential functions; exploring such concepts as the base, growth factor and decay factor; how different values for these terms affect the graph of the function.

Plotting a logarithmic function on the coordinate plane to see how it is the mirror image of a corresponding exponential function; logarithms can be disorienting; studying their properties to discover how they make certain calculations simpler.

Delving deeper into exponential and logarithmic functions with the goal of solving a typical financial investment problem using the "Pert" formula; studying the change of base formula for logarithms; the special function of the base called e.

Pascal's triangle is an array of numbers that corresponds to the coefficients of binomials of different powers; connecting combinatorics with algebra; the formula for each value in Pascal's triangle, the factorial function and the binomial theorem.

Using the factorial function to solve problems in permutations and combinations; figuring out what all the permutations of the letters a, b and c are; how many combinations of four books are possible when one has six to choose from.

A short introduction to probability; celebrating the completion of the series with a deck of cards; finding out if one can use the principles of probability, permutations and combinations to calculate the probability of being dealt different hands.