In this lesson, our instructor Vincent Selhorst-Jones gives an introduction to polynomials. He discusses the polynomial function, polynomial equation, degree, distributive property, behavior or polynomials, and the leading coefficient test.
where n is a nonnegative integer, an, an−1, ..., a0 are all real numbers (constants), and an ≠ 0.
If the above definition is a little hard to understand, here are the key ideas:
It starts from some nonnegative integer n. This number is the exponent that the very first x has: xn.
It has this structure: xn + xn−1 + …+ x2 + x + , where each of the blanks is filled with a number. (That's what all those a's represent.)
The a's (blank spaces above) can potentially be 0's, causing that spot to "disappear". The only spot that's not allowed to be 0 is an: the first spot. This means our xn is not allowed to disappear. (Otherwise why use n if we won't
Putting all this together, we get expressions like x4 + 3x2 − 9x + 17.
A polynomial function is a function that is made from a polynomial, like f(x) = x4 + 3x2 − 9x + 17.
A polynomial equation is an equation made from a polynomial, like y = x4 + 3x2 − 9x + 17.
While we will generally use x as the variable in polynomials, we should note that any variable can be used. Like in our work with functions, x is a commonly used variable, but there are others out there.
The degree of a polynomial is the size of the largest exponent on a variable. If the polynomial isn't in order of largest to smallest exponents, the degree might not necessarily be the first exponent you see.
Some types of polynomials come up often enough that they get special names. Sometimes the name is based on the degree of the polynomial:
Cubic - Degree 3
Quadratic - Degree 2
Linear - Degree 1
Constant - Degree 0
Other times, the name is based on how many terms it has:
Trinomial - 3 Terms
Binomial - 2 Terms
Monomial - 1 Term
Polynomials can often be broken down into multiplicative factors by the distributive property (multiplication over parentheses). Occasionally we want to take two factors and multiply them together to expand the polynomial. In the most basic
form of two binomials, we have the FOIL method:
(a+b) (x+y) = ax + bx + ay + by.
This idea can work on larger factors or more than two as well: each term in a parenthetical group multiplies all the terms in the other parenthetical group.
The reverse of expanding is called factoring, which we will explore extensively in later lessons.
The long-term behavior of a polynomial is determined by the term with the largest exponent and whether or not that term has a positive or negative coefficient. This is called the Leading Coefficient Test. [To visually see what happens, check
out the video for the various images. In general, though, you can imagine what ±x2 and ±x3 would do.]
Intro to Polynomials
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.