Polynomial expressions, equations, & functions

This course is a part of Algebra (all content), a 20-course Topic series from Khan Academy.

This topic covers: - Adding, subtracting, and multiplying polynomial expressions - Factoring polynomial expressions as the product of linear factors - Dividing polynomial expressions - Proving polynomials identities - Solving polynomial equations & finding the zeros of polynomial functions - Graphing polynomial functions - Symmetry of functions

This course contains 37 segments:

Intro to polynomials

Learn about polynomial expressions: What are they? How are they constructed? What can we do with them?

Adding & subtracting polynomials

Learn how to add and subtract polynomial expressions with one variable.

Adding & subtracting polynomials: two variables

Learn how to add and subtract polynomial that involve two variables. For example, x^3 + xy + 3y - (x^3 + 6xy + 2y^2).

Multiplying monomials

As an intro to more elaborate polynomial multiplication, learn how to multiply monomials (which are polynomials with a single term). For example, multiply 2x³ and 5x⁷.

Multiplying monomials by polynomials

Learn how to multiply a polynomial expression by a monomial expression. Monomials are just polynomials with a single term!

Multiplying binomials

Learn how to multiply two binomials together. For example, (3x - 7) * (10x + 2).

Special products of binomials

Learn about the special types of products of binomials: perfect squares and the difference of two squares. These will be very helpful once you tackle more advanced expressions in Algebra.

Multiplying binomials by polynomials

Learn how to multiply a polynomial expression by a binomial expression.

Polynomials word problems

See a few examples of how we can represent real-world situations with polynomials.

Introduction to factorization

Learn what factorization is all about, and warm-up by factoring some monomials.

Factoring monomials

Learn how to write a monomial as a factor of two other monomials. For example, write 12x^3 as (4x)(3x^2).

Factoring polynomials by taking common factors

Learn how to take a common monomial factor out of a polynomial expression. For example, write 2x^3+6x^2+8x as (2x)(x^2+3x+4).

Evaluating expressions with unknown variables

Learn how to evaluate expressions with variables whose values are unknown, by using another information about those variables. For example, given that a+b=3, evaluate 4a+4b.

Factoring quadratics intro

Learn how to factor quadratic expressions with a leading coefficient of 1. For example, factor x²+3x+2 as (x+1)(x+2).

Factoring quadratics by grouping

Learn how to factor quadratic expressions with a leading coefficient other than 1. For example, factor 2x²+7x+3 as (2x+1)(x+3).

Factoring polynomials with quadratic forms

Learn how to factor quadratic polynomials of the form ax^2+bx+c as the product of two linear binomials. For example, write x^2+3x-10 as (x+5)(x-2). Learn how to identify these forms in more elaborate polynomials that aren't necessarily quadratic. For example, write x^4-4x^2-12 as (x^2+2)(x^2-6).

Factoring quadratics: Difference of squares

Learn how to factor quadratics that have the "difference of squares" form. For example, write x²-16 as (x+4)(x-4). Learn how to identify this form in more elaborate expressions. For example, write 4x²-49 as (2x+7)(2x-7).

Factoring quadratics: Perfect squares

Learn how to factor quadratics that have the "perfect square" form. For example, write x²+6x+9 as (x+3)². Learn how to identify these forms in more elaborate expressions. For example, write 4x²+28x+49 as (2x+7)².

Strategy in factoring quadratics

There are a lot of methods to factor quadratics, which apply on different occasions and conditions. Now that we know all of them, let's think strategically about which of them is useful for a given quadratic expression we want to factor.

Factoring polynomials with special product forms

Factor polynomials of various degrees using factorization methods that are based on the special product forms "difference of squares" and "perfect squares." For example, factor 25x⁴-30x²+9 as (5x²-3)².

Long division of polynomials

Learn how to divide two polynomials using long division.

Synthetic division of polynomials

Learn how to divide polynomials using long division.

Practice dividing polynomials with remainders

After learning about the different methods in which we can find the quotient and the remainder of two polynomials, gain some practice with actually performing polynomial division yourself.

Polynomial Remainder Theorem

The polynomial remainder theorem allows us to easily determine whether a linear expression is a factor of a given polynomial. Learn exactly what the theorem means, practice using it, and learn about its proof.

Binomial theorem

Learn how to expand powers of binomial expressions (which are polynomial expressions with exactly two terms). This is done using the binomial theorem!

Understanding the binomial theorem

Now that you know how to use the binomial theorem in order to expand powers of binomial expressions, let's gain further insight into why this actually works!

Advanced polynomial factorization methods

Learn more ways to factor polynomials with degree higher than 2.

Proving polynomial identities

Practice proving polynomial identities, using all the factorization and expansion methods you know.

Polynomial identities with complex numbers

Prove more polynomial identities, this time including complex numbers!

Quadratic equations with complex numbers

Remember all these quadratic equations with "no real solution"? Well, it turns out those equations do have a solution, it's just a complex number! Solve a bunch of those here.

Fundamental Theorem of Algebra

You may already have noticed that quadratic equations always have a solution when including complex number solutions. It turns out this is true for any polynomial equation, of any degree! Learn about this and more, right here.

Finding zeros of polynomials

Use all the knowledge you have about polynomials to find their zeros (which are the input values that make the polynomial equal to zero).

Zeros of polynomials and their graphs

Learn how to use the zeros of polynomials to draw a pretty good sketch of their graphs.

End behavior of polynomial functions

Learn about the end behavior of polynomial functions. End behavior is the way the function behaves as the input values grow infinitely positive or infinitely negative.

Graphs of polynomials

Combine your knowledge about the zeros and the end behavior of a polynomial in order to sketch its graph.

Introduction to symmetry of functions

You may already be familiar with types of symmetries of geometrical shapes. Learn how functions can be symmetrical too!

Symmetry of polynomial functions

Learn how to determine the symmetry of a polynomial function.