Ellie's Blog: January 14th and 15th

Hi Everyone!

This blog is for Monday and Tuesday of this week. 

We started a new topic on Monday: POLYNOMIALS!

What is a polynomial?

"Poly" means many terms, so a polynomial is the sum of many terms. 

Let's look at some examples.

These ARE Polynomials:

• y=2x+3 → Line
• y=3x²-4x+7 → Quadratic
• y=-4/3x¹⁰⁰-75.23x²+4
• y=5x

These Are NOT Polynomials:

• y=5^x → Exponential Function
• y=√3x³ → Square Root
• y= 4/x+1 → Variable in Denominator

How do you form a polynomial?

A polynomial is made up of two parts: A coefficient and an x^exponent

These are multiplied together to form a polynomial. The coefficient must be a real number and the exponent must be an integer, either positive or zero. In other words, no negative exponents. 

Here is the general form of a polynomial:

The "a"s represent the coefficients of the terms

Vocab:

There's some key vocab that's important in understanding polynomials:

Leading Term: the term with the largest exponent

Leading Coefficient: the coefficient of the leading term

Degree: the value of the largest exponent

This Khan Academy video breaks down a polynomial:

https://www.khanacademy.org/math/trigonometry/polynomial_and_rational/polynomial_tutorial/v/terms-coefficients-and-exponents-in-a-polynomial

Then we moved on to graphing. For this section, it is super important to have a graphing calculator....Colby....

First of all, it's good to remember that all graphs of polynomials should be smooth.

We put a list of terms on the board that looked like this:

y=x → linear → odd

y=x² → quadratic/parabola

y=x³ → cubic (3^rd degree) and symmetric over origin (odd)

y=x⁴ → quartic (4^th degree)

y=x⁵ → quintic (5^th degree)

y=x⁶ → sextic (6^th degree)

Then, we graphed each of these

y=x:

y=x²

y=x³ 

y=x⁴

y=x⁵

y=x⁶

^{As you can see, many of these graphs are very similar. Notice that all of the even degrees produce parabolas, where the odd degrees produce the sort of sideways "s" shape (I don't know if it has a name). It's important to see the differences between them though.} 

Monday night's homework was p. 233: 1-5 odd, 7-12 all, 13-19 odd

It was fairly straightforward. The only confusing part was when we were asked to find the "zeros" of the equations. We hadn't gotten to that in class and we went over it more on Tuesday, so that leads me to Tuesday's class. 

Finding the zero just simply means finding the X-intercepts. 

In #17, we are asked to find the zeros of the function, x³ - x. The graph of this function looks like this:

Here we can see that the zeros (x-intercepts) are: -1, 0, 1

You can find the zeros on your graphing calculator by selecting "zero" under the "calc" feature. Under the "calc" feature, you can also find the maximum and minimum of a function. 

Alright, I think that takes care of most of what we talked about during Monday and Tuesday's classes. Chapter 3 in the book is a good place to learn more about polynomials. Let me know if I missed anything or if you have any questions. 

Here's some good math humor!

Sources of images:

http://www.mathsisfun.com/algebra/polynomials-general-form.html