Polynomial Review - Student Explain Everythings

Gleaning X-Intercepts of Functions Using Synthetic Division to Factor

On Thursday and Friday we experimented with using long and synthetic division to find x-intercepts of functions. We used problems from the handout which we were given in class on Tuesday I believe, and I will be referring to problems from it. I will also include it subsequently in this post.

To jog your memories of the remainder theorem, a shortcut which we learned early in the week, I'll go to #29 on the worksheet, the example we used in class.

Since the divisor(bottom term) is (x+3), and -3 is the zero to be derived from that, we plug in -3 as x into the dividend(top term). Don't plug it into the whole equation, just the top of the division. The number you should get(in this case -1) is your remainder.

And since I know that an ineradicable elevated speed of learning is a formidable combatant, I have embedded a video which should explain my feelings on modern organized education's interplay on our thoroughly exacerbated sentiments. If you look closely, you should notice that it is a cat.

Moving on from that, I would like to re-edify the concept of precisely calculating the occasional uncongenial x-intercepts of functions, as well as augment a sense of efficacy that we should all carry with us in our spoken rhetoric when we find ourselves referring to mathematics, as well as assist you in your quest for knowledge concerning the relationships of factoring functions and x-intercepts. To complete both of these tasks at once, assuming you all have read this far and not simply scrolled to the various cat videos which I would neither blame you for nor put past you, I will be using problem 33 from page 154 in our book.

The most intelligible and simple way to find the x-intercepts is by plugging the equation into your graphing calculator, hopefully everyone has theirs fully charged, because we are going to be doing some hardcore mathing.

With a standard window, your graph of x³ + 3x² - 2x - 6 should look like this:

Right away, we could guess that -3 could be an x-intercept, and by simply pressing -3, then hitting "enter", it will confirm our suspicions. Now, we could use our calculators to calculate the other zeros of this function, but it would only give them as decimals, and the ones that remain are not particularly copacetic round numbers. To combat this dilemma, we will divide the original cubic function by the only factor which we now know exists: (x + 3). Using synthetic division on this is the easiest way, so I will gu hold your horses I just found another cat video.

Once you complete the synthetic division of the cubic you should get (x² - 2) as the quotient. The best thing to do with a quadratic, as it turns out, it set it equal to zero. This one is rather easy; no use of the quadratic formula is required. Your answer for x is the other two x-intercepts of the original cubic function.

In addition to using synthetic division to precisely calculate zeros, we also learned what multiplicity and roots of multiplicity are. For an example, in the function f(x) = (x + 5)2(x - 4), a root of multiplicity is -5, one of the x-intercepts. For the multiplicity of -5, look to the exponent of the factor in which it resides. The multiplicity of -5 is 2. The multiplicity of 4 is 1 in this function, but when the multiplicity is 1, we usually don't say it.

To take these concepts further, lets look at problem 64 on the back of the worksheet. To determine an equation for this degree 3 function, look at its x-intercepts. even though it is degree 3, it only has two, -1 and 2. Right from there you can start building your equation. f(x) = (x + 1)(x - 2). But wait, this is supposed to be a cubic, so we need a way to fit one more x in the equation. If you look at the graph, the section where the function touches the x-axis at (2,0) looks nearly parabolic. If you try squaring the (x - 2) part of the equation, you will find that that is the missing piece to this cubic. Your final equation is f(x) = (x + 1)(x - 2)2.

Now I realize that I am currently overdue for another cat video, and knowing that I simply cannot exceed the strangeness posed by Colby "drop it like its hot" Harvey's demonstration video on synthetic division, I will embed another, featuring an old guy someone in the comments named "Gandalf on vacation". For a real treat, go to the statistics of the video, and check out the top demographics.

I think I will let that conclude my blog post, its been a pleasure helping you understa I found another cat video. Dat face.

~Austin

Synthetic & Long Division &other stuff

ayyy guys r u ready 4 sum math bloggin??!!

On Thursday we were introduced to Graphing With Factored Polynomials

y= (x-2)(x+7)(x-5) --> Degree 3 because there are 3 x's, also meaning that it will cross the x axis 3 times.
so then we found that in this problem it crossed the x axis at (-7, 0), (2, 0), (5, 0) and we were all like "daanngg its so easy to see the x-intercepts when the problem is factored!!!!!"
But just to make sure everyone remembers why we thought it was so easy, lets do it out.

(x-2)  - then we ask ourselves "hmm..what do i plug in for x to make this come out 0? OH A 2!!! "and thats how you got (2, 0).

(x+7) "hmmmm...how do i get to 0???" OH a -7!! --> (-7, 0)

(x-5) "Oh I get it, I get it, a 5 would make it zero.Wow, this is so much easier when factored!"--> (5, 0)

My x-intercepts are (2, 0), (-7, 0), and (5, 0) because when I plug them in I get 0.

Next in class we went up to the board and drew some examples:

#1

#2

#3

Then we were introduced to Long Division.

1st we recalled doing long division with numbers.

This is just a general idea of what your answer should look like. The quotient is what you get after you do the long division

And if you're having a hard time remembering how to do long division with numbers here's a video/song that will help you understand how to do it in a catchy way.

http://www.youtube.com/watch?v=R_cqrdZNmr0

Then we tied long division in with polynomials.

Here's an example:

1. x times what gives you x^2 (put that answer on top of the dividing bar.)

2. multiple the answer you got with the leading term on the outside of the dividing bar. (x time x) **You should always get the same as the first term on the inside of the bar, so they cancel out**

3. 1 times x^2 = x bring that answer down next to the x^2

4. subtract

5. repeat with other terms of the dividend

Another example:

That was the end of Thursday's class.

Friday, we reviewed a lot of long division but then we were introduced to Synthetic Division.

**** Synthetic Division only works when you are dividing by x-c [x+c = x-(-c)] ****

two key things to remember during synthetic division are:

1.write down coefficients.

2. flip the sign of the divider.

http://www.youtube.com/watch?v=fdUQuQ-AYM4

Hopefully that video helped.

Here's one last example:

ok thxxx c u thursday!! (cuz we gon get a snowday 2morro) pce out gurlz 'n boiz

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

I guess as a second semester senior I am off to a good start having to blog second after Nate.  So here goes:

I'm sorry, having permission to post videos is just an invitation for a cat video.

Thursday I come to class so nervous I will miss something and not be able to talk about it in the blog, so I took really good notes right from the start. This is what I have down,

Thursday: First blog day. 
-Talked alot about the blog
-Nate set the bar really high
-Talking about what to comment on

Going over homework....

Page136 59, 63, 65, 67, 70-73

Question on 67.

y=sq.root of x

Then, we plugged in different numbers for x, like 0, 4, (and I said 16, but Lisa didn't want to draw that many lines on the graph) and 9. 

That gave us a graph that looked like this: 

New function --> y= sq. root -x shifts over y axis
                        y= sq.root -(x+2) moves it left 2
                        y= sqroot -(x+2)-1  down 1

Then we graphed it on the calculator to look at it. 





Well, that is all I wrote down. The rest of class was spent working on a worksheet. 

We were given f(x) or g(x) and had to sketch each transformation on the same coordinate axes.

They were pretty straight forward, but most of them took many steps.  

I found 8b to be annoying. It had four steps total.  

8b) -1/2f(x-1) + 5

So.. this will be difficult because I can't decide if I want to upload this graph with my answer on it or just look at the transformation...Let's look at the transformation first.

 -1/2 f (x-1) + 5 Flip over x-axis 
-1/2 f (x-1) + 5 Divide by 2
-1/2 f (x-1) + 5 Shift right 1
-1/2 f (x-1) + 5 Shift up 5

*Remember, the -1 goes the opposite direction, +1, but the +5 outside the parentheses doesn't switch 


Here is the final picture (my line looks off, not sure why, but hopefully it is close to the right shape)

So Colby wasn't lying, Big Comfy Couch is a real show.

And that is all we did in class in class on Thursday the 10th.


On Friday, we didn't do much.  We went over our homework which was on page 136 and #76-83 and 86-89.

Most of class was spent working on our homework which is a worksheet of graded problems.  The sheet covered symmetry and transformations.  I'll show a couple examples from the sheet, but Friday's class was mostly wrapping up this unit.  

On the handout:

25) Sketch a graph that is symmetric with respect to both the x-axis and the y-axis

26) Sketch a graph of a function that is symmetric with respect to the origin

28) Determine whether the graph of the given equation is symmetric with respect to the x-axis, y-axis or the origin.

5y=7x2-2x

y-axis Test: 5y=7(-x)2-2(-x)
5y=7x2+2x
No

x-axis Test: 5(-y)=7x2-2x
-5y=7x2-2x
No

Origin Test: 5(-y)=7(-x)2-2(-x)
-5y=7x2+2x
No


As a reminder of the tests:

The y-axis test: plug in -x for all x's

The x-axis test: plug in -y for all y's

The origin Test: plug in both -y and -x


I didn't get very far in the homework in class, but if anyone has any trouble with it, page 153 in the book was very helpful!
And this is now the end of my blog.

   

Nate Hansen 01.07.13. & 01.08.13. Blog Post.

Hey guys, so I am blogging for Monday and Tuesday of this week.


Monday: Today we mainly went over the homework that was assigned from Friday {p. 135 #s 1-6, 11, 15, 21, 27-32}. The homework assignment was focused on identifying x-axis symmetry, y-axis symmetry, or symmetry with respect to the origin. Here are some examples:

 

URL: http://tutorial.math.lamar.edu/Classes/Alg/Symmetry_files/image003.gif

Symmetric with respect to the x-axis

As you will remember, an easy visual way to test for x-axis symmetry is to imagine folding the graph along the x-axis. If the parts above and below the x-axis, coincide, the graph is symmetric with respect to the x-axis.

The algebraic test for x-axis symmetry is as follows: replace (y) with (-y). If it comes out to an equivalent equation, then the graph is symmetric to the x-axis.

 

 

URL: http://03.edu-cdn.com/files/static/mcgrawhill-images/9780071439275/f0056-02.jpg

Symmetric with respect to the y-axis

As with the x-axis; in order to find out whether or not a graph represents y-axis symmetry, you can mentally picture folding the graph along the y-axis. If the components of the graph to the right and left of the y-axis match up, the graph is symmetric to the y-axis. 

The algebraic test for y-axis symmetry is almost exactly like the test for x-axis symmetry: replace (x) with (-x). If it comes out to an equivalent equation, then the graph is symmetric to the y-axis.

Remember: If the graph of a function f is symmetric with respect to the y-axis, it is called an even function. Mathematically speaking, for each x in the domain of f; f(x) = f(-x).

 

URL: http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/typesofsymmetry_clip_image029.gif 

 Symmetric with respect to the origin

In order to easily determine whether a graph represents symmetry with respect to the origin, mentally rotate the graph 180˚ about the origin. The resulting figure should coincide with the original graph.

The algebraic method to test for symmetry around the origin is to first replace (x) with (-x) and (y) with (-y). After this, and it comes out to an equivalent equation as your original equation, then the graph is symmetric with respect to the origin.

Remember: If a graph of a function f is symmetric about the origin, we say that it is an odd function. Mathematically speaking, for each x in the domain of f; f(-x) = -f(x).

I'll do problem #17 on page 135 to illustrate the points made above.

Test algebraically whether the graph is symmetric with respect to the x-axis, the y-axis, and the origin. Then check your work graphically, if possible, using a grapher.

17. 3x² – 2y² = 3

This is your baseline equation to which you should compare other equations when you have inserted either (-x), (-y), or both.

Plug in (-y) and simplify: 3x² - 2(-y)²  = 3

3x² – 2y² = 3

This is the same as the baseline equation, so it is symmetric with respect to the x-axis.

Next plug in (-x) and simplify to see if the graph is also symmetric about the y-xis: 3(-x)² – 2y² = 3

3x² – 2y² = 3

This is also equivalent to the original equation, so the graph is also symmetric with respect to the y-axis.

Finally plug in both (-y) and (-x) and simplify to test for symmetry around the origin: 3(-x)² – 2y² = 3

3x² – 2y² = 3

This too is the same as the initial equation so the graph is symmetric to the origin as well as to the x- and y- axes.

All of the material for Monday's class that I blogged about either came from notes that I took from the board, or from pages 122-127 in our Precalculus textbook. Also see the homework assignment mentioned in the first part of this post. If you are still having trouble understanding the concepts of different symmetries and the even vs odd idea, then check out this Khan Academy video, it is pretty helpful: 

https://www.khanacademy.org/math/trigonometry/functions_and_graphs/analyzing_functions/v/recognizing-odd-and-even-functions

That essentially sums up class on Monday: mostly reviewing homework and concepts that we studied during the previous Friday's class. Now we'll move on to talk about Tuesday. 

Tuesday: Today we reviewed homework from the previous night {p.135 # 31-44, and #7 on worksheet: a,b,c,},  and learned more about the different transformations that a function can undergo and how to recognize such a transformation. Go to page 134 in our book to find the following helpful chart:

A

Again, you can find the above chart on page 134 of the textbook.

Using the knowledge from the above table, I will now show problem #63 on page 136.

Write an equation for a function that has a graph with the given characteristics. Check your answer using a grapher. 

63. The shape of y = 1/x but shrunk vertically by a factor of ½ and shifted down 3 units

First put the equation in to function form as follows: f(x) = 1/x

Next multiply the side of the equation with “1/x” by 2 in order to shrink the function. Refer back to the chart above if you need help remembering other transformations.

Your function should now look like this: f(x) = 1/2x

Finally, subtract 3 from the “1/x” side of the equation to vertically shift the function downwards three units.

Your final function will look like: f(x) = 1/2x -3

I will do one more problem, problem #65 on page 136. The instructions are the same as problem # 63

65. The shape of y = x², but upside-down and shifted right 3 units and up 4 units

As with the previous problem, convert the equation into function form: f(x) = x²

Make the x term negative to flip the function across the x-axis

The function will look like: f(x) = -x²

Now, in order to account for shifting the function right 3 units, subtract three from the x term.

The function will now look like: f(x) = -(x – 3)²

Finally, add 4 to the right side of the equation to vertically shift the function up four units

This is the final form of the transformed function: f(x) = -(x – 3)² + 4

That's it for my blog post. Hope it helps clear some stuff up. Thanks. 

Nate

It's 2013, and we're blogging!!

I'm very excited about the ideas, questions, concepts, debates, theories, etc. you will all post on this site this semester!  For inspiration, take a moment to view some of the best math scribe blog posts in The Scribe Post Hall of Fame!