Unit 3 of 6

Advanced Math

  • 18 lessons
  • Parts 39 to 74

Advanced Math covers the nonlinear algebra on the test. This unit works with exponents, radicals and polynomials, factors expressions by recognizing their structure, and solves quadratic equations by factoring, completing the square and the quadratic formula.

The second half reads quadratic and polynomial graphs, solves rational, radical and absolute value equations, intersects lines with curves, models growth and decay with exponential functions, and shifts and stretches graphs.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Powers and Expressions

  1. Exponent Rules

    How the rules for products, quotients, powers, zero and negative exponents follow from counting repeated factors, and which values of the base each rule must exclude.

    Theory part 39Practice part 40

  2. Radicals and Rational Exponents

    How square roots, cube roots and fractional exponents describe the same numbers, why even roots need nonnegative inputs, and why the square root of x squared is the absolute value of x.

    Theory part 41Practice part 42

  3. Expanding and Combining Polynomials

    How to multiply out products of polynomials without losing a term, combine only like terms, and match coefficients to find unknown constants in equivalent expressions.

    Theory part 43Practice part 44

  4. Factoring and Recognizing Structure

    How to rewrite a polynomial as a product using common factors, trinomial patterns, special products and grouping, and how to confirm any factorization by expanding it back.

    Theory part 45Practice part 46

Quadratics

  1. Solving Quadratic Equations by Factoring

    How to solve a quadratic equation by moving everything to one side, factoring, and setting each factor equal to zero, and why dividing by a variable can silently lose a solution.

    Theory part 47Practice part 48

  2. Completing the Square

    How to add the right number to both sides so a quadratic becomes a perfect square, then solve it with both square roots or read its vertex.

    Theory part 49Practice part 50

  3. Quadratic Formula and Discriminant

    How to solve any quadratic equation with the quadratic formula, and how the discriminant tells you whether there are two, one or no real solutions before you solve.

    Theory part 51Practice part 52

  4. Quadratic Graphs and Their Forms

    How to read a parabola's vertex, axis of symmetry and intercepts, and how to pick the standard, factored or vertex form that shows the feature a question asks about.

    Theory part 53Practice part 54

  5. Quadratic Parameters and Root Relationships

    How to turn facts about a quadratic's solutions, such as one known root, a repeated root or no real roots, into an equation for an unknown constant, and how to avoid the case where the equation stops being quadratic.

    Theory part 55Practice part 56

Nonlinear Equations and Functions

  1. Polynomial Zeros and Graph Features

    How a factored polynomial shows its real zeros, how the sign of each factor decides where the graph is above or below the axis, and what a repeated factor does to the graph near a zero.

    Theory part 57Practice part 58

  2. Rational Expressions and Excluded Values

    How to simplify, add, subtract, multiply and divide rational expressions by working with factors, while keeping every value that made an original denominator zero out of the domain.

    Theory part 59Practice part 60

  3. Solving Rational Equations

    How to solve an equation with variables in denominators by first noting the excluded values, then clearing every denominator, and finally checking each candidate in the original equation.

    Theory part 61Practice part 62

  4. Solving Radical Equations

    How to solve an equation with a square root by isolating the root and squaring, why squaring can produce a candidate that does not work, and how to check every candidate in the original equation.

    Theory part 63Practice part 64

  5. Absolute Value Equations and Graphs

    How absolute value measures distance, why most absolute value equations split into two cases, when a case must be thrown out, and how the V-shaped graph shows the number of solutions.

    Theory part 65Practice part 66

  6. Systems with Lines and Curves

    How to find every point a line shares with a parabola or a reciprocal curve, how to tell two, one or no meeting points apart, and why answers are ordered pairs.

    Theory part 67Practice part 68

  7. Exponential Growth and Decay

    How to read the initial value, the factor and the time unit in an exponential model, rewrite it for a different time interval, and solve for an unknown exponent with a common base.

    Theory part 69Practice part 70

  8. Shifting, Reflecting and Stretching Graphs

    How a change in a function rule moves each point of its graph, why horizontal changes run through the input, and how to carry the domain along for quadratic, exponential, cubic and reciprocal graphs.

    Theory part 71Practice part 72

  9. Linear and Exponential Models from Tables

    How to tell from a table whether a quantity grows by a constant difference or a constant ratio over equal input steps, and how to turn that pattern into a linear or exponential model.

    Theory part 73Practice part 74