Unit 2 of 10

Polynomial, Rational, and Algebraic Functions

  • 18 lessons
  • Parts 35 to 70

Polynomials and quotients of polynomials are the functions you can compute with addition, multiplication and division alone, which makes them the first models to reach for. This unit starts with quadratics, circles and root functions, then studies polynomials through their end behavior, zeros and factors.

Complex numbers enter so that every polynomial can be factored completely, as the fundamental theorem of algebra promises. The second half turns to rational functions, with their holes and their vertical, horizontal and slant asymptotes, and finishes with polynomial and rational inequalities solved by sign charts.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Quadratics, Circles, and Roots

  1. Quadratic Functions and Vertex Form

    How to find the vertex, axis and zeros of a quadratic function by completing the square, factoring or the quadratic formula, and how the discriminant and the setting of a problem decide which answers count.

    Theory part 35Practice part 36

  2. Circles, Distance, and Midpoints

    How the distance and midpoint formulas follow from a right triangle, how they give the equation of a circle, and how completing the square finds the center and radius from an expanded equation.

    Theory part 37Practice part 38

  3. Power Functions and Root Functions

    How integer and rational exponents are defined, which inputs each power and root function accepts, how roots undo powers, and why the square root of x squared is the absolute value of x.

    Theory part 39Practice part 40

  4. Solving Rational and Radical Equations

    How to clear fractions and remove square roots without losing track of the domain, why these steps can create false solutions, and how to test every candidate in the original equation.

    Theory part 41Practice part 42

Polynomial Functions

  1. End Behavior of Polynomials

    How the degree and the sign of the leading coefficient decide what a polynomial does far to the left and far to the right, and why the leading term takes over for large inputs.

    Theory part 43Practice part 44

  2. Zeros and Multiplicity of Polynomials

    What zeros, roots and x-intercepts of a polynomial are, how the multiplicity of a zero decides whether the graph crosses or touches the axis there, and why some zeros never appear on the graph.

    Theory part 45Practice part 46

  3. Polynomial Division

    How to divide one polynomial by another with long division or synthetic division, what the quotient and remainder mean, and how to check a division by multiplying back.

    Theory part 47Practice part 48

  4. The Remainder and Factor Theorems

    Why dividing a polynomial by x minus c leaves the value of the polynomial at c as the remainder, how that turns factoring into checking a single value, and how to split off linear factors.

    Theory part 49Practice part 50

  5. Rational Zeros Theorem and Factoring

    How the coefficients of a polynomial with integer coefficients limit its possible rational zeros to a short list, how to test that list quickly, and why the list can miss irrational and complex zeros.

    Theory part 51Practice part 52

Complex Numbers and Complete Factorization

  1. Complex Numbers and the Complex Plane

    What the imaginary unit is, how to add, multiply and divide numbers of the form a + bi, how to picture them as points in a plane, and how they give every quadratic equation its solutions.

    Theory part 53Practice part 54

  2. The Fundamental Theorem of Algebra

    Why every nonconstant polynomial has a complex zero, why a polynomial of degree n has exactly n zeros counted with multiplicity, and how real polynomials split into real linear and quadratic factors.

    Theory part 55Practice part 56

Rational Functions

  1. Rational Functions and Domain Restrictions

    What a rational function is, how its domain comes from the original denominator, and why cancelling a common factor keeps every excluded value, often leaving a hole in the graph.

    Theory part 57Practice part 58

  2. Vertical Asymptotes and Holes

    How to tell whether an excluded input of a rational function gives a vertical asymptote or a hole, and how signs show which way the graph runs on each side of an asymptote.

    Theory part 59Practice part 60

  3. Horizontal Asymptotes

    How the leading terms and a comparison of degrees decide whether a rational function levels off far out, at which height it does so, and why the graph may still cross that line.

    Theory part 61Practice part 62

  4. Slant (Oblique) Asymptotes

    How polynomial division finds the tilted line that a rational function follows far out when the top has degree one more than the bottom, and why the leftover fraction fades to zero.

    Theory part 63Practice part 64

  5. Graphing Rational Functions

    How to combine the domain, holes, intercepts, signs on test intervals and the asymptotes into one accurate sketch of a rational function, including where it crosses a horizontal asymptote.

    Theory part 65Practice part 66

Polynomial and Rational Inequalities

  1. Polynomial Inequalities

    How to solve a polynomial inequality by finding its critical values, testing one point in each interval of a sign chart, and writing the solution set with the right endpoints.

    Theory part 67Practice part 68

  2. Rational Inequalities

    How to solve an inequality with a variable in a denominator by comparing one fraction with zero, cutting the line at zero points and undefined points, and testing each critical interval.

    Theory part 69Practice part 70