Trigonometry and precalculus
Practice this topic
Polynomial and rational functions
Exploring quadratics and higher degree polynomials. Also in-depth look at rational functions.
Factoring quadratics
Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer.
This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!
- Factoring quadratic expressions
- Examples: Factoring simple quadratics
- Factoring polynomials 1
- Example 1: Solving a quadratic equation by factoring
- Example 2: Solving a quadratic equation by factoring
- Solving quadratics by factoring
- Example 1: Factoring trinomials with a common factor
- Factoring polynomials 2
- Solving quadratics by factoring 2
- Factoring Special Products
- Example 1: Factoring difference of squares
- Example 2: Factoring difference of squares
- Factoring difference of squares 1
- Factoring difference of squares 2
- Factoring difference of squares 3
- Example 5: Factoring by grouping
- Example 6: Factoring by grouping
- Factoring polynomials by grouping
Completing the square and the quadratic formula
You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula.
Welcome to the world of completing the square!
- Solving Quadratic Equations by Square Roots
- Solving quadratics by taking the square root
- Solving Quadratic Equations by Completing the Square
- Completing the square 1
- Completing the square 2
- How to Use the Quadratic Formula
- Proof of Quadratic Formula
- Quadratic formula
- Example: Complex roots for a quadratic
- Discriminant of Quadratic Equations
- Discriminant for Types of Solutions for a Quadratic
- Solutions to quadratic equations
Graphing quadratics
Tired of lines? Not sure if a parabola is a disease of the gut or a new mode of transportation? Ever wondered what would happen to the graph of a function if you stuck an x² someplace? Well, look no further.
In this tutorial, we will study the graphs of quadratic functions (parabolas), including their foci and whatever the plural of directrix is.
- Ex 3: Graphing a quadratic function
- Example: Roots and vertex of a parabola
- Example: Parabola vertex and axis of symmetry
- Graphs of Quadratic Functions
- Graphing parabolas in standard form
- Parabola Focus and Directrix 1
- Focus and Directrix of a Parabola 2
- Vertex of a parabola
- Graphing parabolas in vertex form
- Graphing parabolas in all forms
- Parabola intuition 3
Quadratic inequalities
You are familiar with factoring quadratic expressions and solving quadratic equations. Well, as you might guess, not everything in life has to be equal.
In this short tutorial we will look at quadratic inequalities.
Polynomials
"Polynomials" sound like a fancy word, but you just have to break down the root words. "Poly" means "many". So we're just talking about "many nomials" and everyone knows what a "nomial" is. Okay, most of us don't. Well, a polynomials has "many" terms.
From understanding what a "term" is to basic simplification, addition and subtraction of polynomials, this tutorial will get you very familiar with the world of many "nomials." :)
- Terms coefficients and exponents in a polynomial
- Addition and Subtraction of Polynomials
- Adding and subtracting polynomials
- Multiplying Polynomials
- Multiplying Polynomials 3
- Multiplying polynomials
- Polynomial Division
- Dividing polynomials 1
- Dividing polynomials with remainders
- Synthetic Division
- Synthetic Division Example 2
- Why Synthetic Division Works
- Factoring Sum of Cubes
- Difference of Cubes Factoring
Binomial theorem
You can keep taking the powers of a binomial by hand, but, as we'll see in this tutorial, there is a much more elegant way to do it using the binomial theorem and/or Pascal's Triangle.
Simplifying rational expressions
You get a rational expression when you divide one polynomial by another. If you have a good understanding of factoring quadratics, you'll be able to apply this skill here to help realize where a rational expression may not be defined and how we can go about simplifying it.
Rational functions
Have you ever wondered what would happen if you divide one polynomial by another? What if you set that equal to something else? Would it be as unbelievably epic as you suspect it would be?
- Adding and Subtracting Rational Expressions
- Adding and Subtracting Rational Expressions 2
- Adding and Subtracting Rational Expressions 3
- Rational Equations
- Solving Rational Equations 1
- Solving Rational Equations 2
- Solving Rational Equations 3
- Subtracting Rational Expressions
- Applying Rational Equations 1
- Applying Rational Equations 2
- Applying Rational Equations 3
- Multiplying and Simplifying Rational Expressions
- Multiplying and Dividing Rational Expressions 1
- Multiplying and Dividing Rational Expressions 2
- Multiplying and Dividing Rational Expressions 3
- Extraneous Solutions to Rational Equations
- Rational Inequalities
- Rational Inequalities 2
Partial fraction expansion
If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions.
This has many uses throughout mathematics. In particular, it is key when taking inverse Laplace transforms in differential equations (which you'll take, and rock, after calculus).