Algebra II (Intermediate Algebra)

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?

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?

Pascal's triangle is a famous triangular array of numbers that corresponds to the coefficients of binomials of different powers. In an episode connecting a branch of mathematics called combinatorics with algebra, investigate the formula for each value in Pascal's triangle, the factorial function, and the binomial theorem.

Delve deeper into exponential and logarithmic functions with the goal of solving a typical financial investment problem using the "Pert" formula. To prepare, study the change of base formula for logarithms and the special function of the base called e.

Plot a logarithmic function on the coordinate plane to see how it is the mirror image of a corresponding exponential function. Just like a mirror image, logarithms can be disorienting at first; but by studying their properties you will discover how they make certain calculations much simpler.

Exponential functions are important in real-world applications involving growth and decay rates, such as compound interest and depreciation. Experiment with simple exponential functions, exploring such concepts as the base, growth factor, and decay factor, and how different values for these terms affect the graph of the function.

Now that you know how to add rational expressions, try the opposite procedure of splitting a more complicated rational expression into its component parts. Called partial fraction decomposition, this approach is a topic in introductory calculus and is used for solving a wide range of more advanced math problems.

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.

Shift your focus to graphs of rational functions, which are the ratio of two polynomials. These graphs are more complicated than those from the previous episode, but their general characteristics can be quickly determined by calculating the domain, the x- and y-intercepts, and the vertical and horizontal asymptotes.

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.

Drawing on your experience with roots and radicals from the previous episode, try your hand at solving equations with these expressions. Begin by learning how to manipulate rational, or fractional, exponents.

Shift gears away from polynomials to focus on expressions involving roots, including square roots, cube roots, and roots of higher degrees (all known as radical expressions). Practice multiplying, dividing, adding, and subtracting a wide variety of radical expressions.

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

Going beyond the approaches you've learned so far, discover how to solve polynomial equations by applying two powerful tools for finding rational roots: the rational roots theorem and the factor theorem. Both will prove very useful in succeeding lessons.

Learn how to solve polynomial equations where the degree is greater than two by turning them into expressions you already know how to handle. Your "toolbox" includes techniques called the difference of two squares, the difference of two cubes, and the sum of two cubes.

Switch from graphs to the algebraic side of polynomial functions, learning how to combine them in many different ways, including addition, subtraction, multiplication, and even long division, which is easier than it seems. Discover which of these operations produce new polynomials and which do not.

Deepen your insight into polynomial functions by graphing them to see how they differ from non-polynomials. Then learn how the general shape of the graph can be predicted from the highest exponent of the polynomial, known as its degree.

Pause to examine the nature of polynomials: a class of algebraic expressions that you've been working with since the beginning of the series. Professor Sellers introduces several useful concepts, such as the standard form of polynomials and their degree, domain, range, and leading coefficients.

Investigate the algebraic properties of the other two conic sections: ellipses and circles. Ellipses resemble stretched circles and are defined by their major and minor axes, whose ratio determines the ellipses' eccentricity.

Delve into the algebra of conic sections, which are the cross-sectional shapes produced by slicing a cone at different angles. In this episode, study parabolas and hyperbolas, which differ in how many variable terms are squared in each.

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.

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

Turn a quadratic equation into an easily solvable form that includes a perfect square, a technique called completing the square. An important benefit of this approach is that the rewritten form gives the coordinates for the vertex of the parabola represented by the equation.

This video concludes the series on solving quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over four slightly more challenging examples where the coefficient of the quadratic term is greater than 1.

This video continues covering the proper way to solve quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over two examples where the coefficient of the quadratic term is greater than 1.

This video continues covering the proper way to solve quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over an example where the coefficient of the quadratic term is greater than 1.

This video continues covering the proper way to solve quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over an example where the coefficient of the quadratic term is greater than 1.

This video continues covering the proper way to solve quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over 5 slightly harder examples that require algebraic manipulation, collecting like terms and expanding terms before proceeding with the factoring step.

This video continues covering the proper way to solve quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over 2 examples where the constant of the quadratic equation is negative.

This video is an introduction to solving quadratic equations of the form ax^2+bx+c=0 by factoring. This video goes over 2 examples modeling the proper way to factor quadratic trinomials that are factorable over the integers.

This video continues covering the general procedure in solving quadratic equations by factoring. This video goes over 3 slightly harder examples modeling the proper way to factor quadratic equations of the form ax^2+bx=0.

This video continues covering the general procedure in solving quadratic equations by factoring. This video goes over 3 examples modeling the proper way to factor quadratic equations of the form ax^2+bx=0 and using the zero product property as a basis to solve quadratic equations of this particular form.

This video is an introduction to solving quadratic equations by factoring. This video goes over 6 examples modeling the proper way to use the zero product property as a basis to solve quadratic equations by factoring.

This video goes over 7 examples covering the proper way to solve quadratic equations of the form a(px+q)^2=0 and a(px+q)^2+c=0, the common method used to solve quadratic equations of these forms is by taking the square root.

This video goes over 5 examples covering the basic notation, concepts, graphs and algorithms to solve quadratic monomials of the form ax^2=0, and quadratic binomials of the form ax^2+c=0.

This video goes over the basic theory and terminology for the purpose of solving quadratic equations. Topics covered include: types of quadratic equations, the standard form of a quadratic equation, and solutions of quadratic equations.