James A Sellers

Professor Sellers explains the topics covered in the series, the importance of algebra, and how you can get the most out of these episodes. Then, launch into the fundamentals of algebra by reviewing the order of operations and trying your hand at several problems.

Continue your study of the link between combinatorics and algebra by using the factorial function to solve problems in permutations and combinations. For example, what are all the permutations of the letters a, b, c? And how many combinations of four books are possible when you have six to choose from?

After a short introduction to probability, celebrate your completion of the series with a deck of cards. Can you use the principles of probability, permutations, and combinations to calculate the probability of being dealt different hands? As with the rest of algebra, once you know the rules, it's simplicity itself!

Extending the exercises on inequalities from a previous episode, step into the realm of quadratic inequalities, where the boundary graph is not a straight line but a parabola. Use your skills analyzing quadratic expressions to sketch graphs quickly and solve systems of quadratic inequalities.

The square root approach to solving quadratic equations works not just for perfect squares, such as 3 × 3 = 9, but also for values that don't seem to involve squares at all. Probe the idea behind this technique, and also venture into the strange world of complex numbers.

Functions are crucially important not only for algebra, but for precalculus, calculus, and higher mathematics. Learn the definition of a function, the notation, and associated concepts such as domain and range. Then try out the vertical line test for determining whether a given curve is a graph of a function.

Combine rational functions using addition, subtraction, multiplication, division, and composition. The trick is to start each problem by putting the expressions in factored form, which makes the calculations go more smoothly. Leaving the answer in factored form also allows other operations, such as graphing, to be easily performed.

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

Using graph paper, experiment with curves formed by simple radical functions. First, determine the domain of the function, which tells you the general location of the graph on the coordinate plane. Then, investigate how different terms in the function alter the graph in predictable ways.

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