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
This is a great post, nice work. It's nice to see the synthetic division done out clearly and neatly. I think you should've included a graph of the cubic function for problem 64 on the worksheet, but besides that I have no other suggestions.
ReplyDeleteAustin, there is lots of good information in your blog post, and lots of cat videos (although, I will admit, they were more mild than I was expecting). You work through some insightful examples, but it might be nice to give the original example in each case instead of referencing a book page or worksheet. Without knowing the original problem, they were hard for me to follow. I like that you included the picture of the graphing calculator screen as well as the neatly done out synthetic division example - both of these enhanced the process you described. Finally, it was great that you covered roots of multiplicity, but a graph of one of your two examples would have been useful so we could know what to look for graphically. Overall, nice post Austin.
ReplyDeleteAustin: This post has really helped me learn more about the process of obtaining the X-intercepts using synthetic division. Sometimes your language seemed a little confusing though. Overall, great job!
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