Solving Polynomial Inequality by Mapping

Today let’s tackle polynomial inequality by curve mapping. We don’t need to test the intervals to check if it satisfies the polynomial inequalities. This could be the fastest way for me to solve this type of problem. Before we start to discuss the method it is important for us to know some terms to be used here.

Important Points

Turning points – In a given polynomial equation with degree n the number of turning points is n-1

Trend – like linear functions, polynomial functions has trend as well.

Given a polynomial function f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+ax+c

If n is even and a_n is positive, both ends of the curve is facing upward.polynomial1

If n is even and a_n is negative, both ends of the curve is facing downward.polynomial2

If n is odd and a_n is positive the trend of the graph is increasing. If the graph is increasing, the graph can be traced from quadrant 3 up to quadrant 1.polynomial3

If n is odd and a_n is negative the trend of the graph is decreasing.If the graph is decreasing, the graph can be traced from quadrant 4 up to quadrant 2.polynomial4

To illustrate the mapping method, let’s provide an example.

Problem:

Solve the solution set of inequality (x-1)(x+2)(x-3)>0

1. Solve the real roots of equation.

The roots are: \{ 1,-2,3\}

2. Draft the roots in coordinate system. The points are (1,0),(-2,0),(3,0)tracing1

3. Take note of the four important points above upon tracing the curve.tracing2

4. The inequality symbol “>” means that the line is above x-axis. The red lines on the curve are the intervals in which the curve is greater than 0 or above x-axis.tracing3

 Worked Problem 1:

Find the solution set of the inequality x^3-4x^2-5x-2\le 0

[toggle title=”Solution:”]

Solve the roots of equation as is, don’t mind the inequality symbol this time. Using synthetic division we have x={2,1(double root)}

Repeat the process mentioned above.

Here is the graph of equation. The solution set is (-\infty,2]question1poly [/toggle]

 

Worked Problem 2:

For what values of x such that the graph of the function f(x)=x^3-4x^2-x+2 below x-axis.

[toggle title=”Solution:”]

The roots of equation are \{-1,1,2\}. F(x)<0  if  (-\infty,-1)\cup(1,2) [/toggle]

Worked Problem 3:

For what values of x does the graph of quartic f(x)=(x-1)^2(x-2)^2 above x-axis?

[toggle title=”Solution:”]

Above x-axis means that y>0 making the equation (x-1)^2(x-2)^2 >0. Since the leading coefficient is greater than 0. Two ends of the graph should be facing upward. The graph also tangent to x-axis through points (1,0) and (2,0) with 3 turning points. Draw a quick draft of the curve and check the values of x in which the y>0. By inspection, y>0 in all real values of x except 1 and 2.

lastxx

 

[/toggle]

 

 

 

 

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

Latest posts by Dan (see all)

You may also like...

116 Responses

  1. Andre says:

    Your place is valueble for me. Thanks!…

  2. hey there and thank you for your info – I have definitely picked up anything new from right here. I did however expertise some technical points using this web site, as I experienced to reload the web site a lot of times previous to I could get it to load correctly. I had been wondering if your web hosting is OK? Not that I am complaining, but slow loading instances times will often affect your placement in google and can damage your high-quality score if advertising and marketing with Adwords. Well I am adding this RSS to my e-mail and could look out for a lot more of your respective fascinating content. Ensure that you update this again soon..

  3. lucy ann says:

    tS5fzs Thanks a lot for the blog.Much thanks again.

  4. this site says:

    GUcBRt Nice info! Also visit my site about Clomid challenge test

  5. Enjoyed every bit of your blog article.Thanks Again. Want more.

  6. Dramacool says:

    This excellent website certainly has all the information and facts I needed about this subject and didn at know who to ask.

  7. Very good article.Thanks Again. Cool.

  8. Wow, incredible blog structure! How lengthy have you been running a blog for? you made blogging glance easy. The full glance of your site is great, let alone the content!

  9. Im thankful for the blog article. Will read on…

  10. I saw someone talking about this on Tumblr and it linked to

  11. This blog is no doubt awesome additionally diverting. I have found helluva helpful stuff out of this amazing blog. I ad love to go back over and over again. Thanks!

  12. Im obliged for the article post. Fantastic.

  13. Thanks for the post.Really looking forward to read more. Fantastic.

  14. Wow! Thank you! I permanently needed to write on my blog something like that. Can I implement a part of your post to my blog?

  15. candle says:

    wow, awesome blog.Really looking forward to read more. Awesome.

  16. UoN says:

    Some genuinely nice stuff on this site, I like it.

  17. Wow, awesome blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your web site is great, as well as the content!. Thanks For Your article about sex.

  18. I truly appreciate this post. I ave been looking everywhere for this! Thank goodness I found it on Google. You have made my day! Thank you again

  19. Superb, what a website it is! This weblog provides valuable information

  20. This is one awesome article. Really Great.

  21. I truly appreciate this article post.Thanks Again.

  22. Thanks-a-mundo for the article post.Thanks Again. Want more.

  23. Only wanna state that this is extremely helpful, Thanks for taking your time to write this.

  24. Really informative blog post.Really looking forward to read more. Cool.

  25. Pretty! This has been an incredibly wonderful post. Thank you for supplying these details.

  26. amiclubwear says:

    Thanks again for the blog post.Much thanks again.

  27. Thank you for the auspicious writeup. It in fact was a amusement account it. Look advanced to more added agreeable from you! By the way, how can we communicate?

  28. I really liked your blog. Really Great.

  29. luggage says:

    This is one awesome blog article. Keep writing.

  30. Really informative article post. Awesome.

  31. Source says:

    Pretty nice post. I just stumbled upon your weblog and wished to say that I ave really enjoyed browsing your blog posts. After all I all be subscribing to your feed and I hope you write again soon!

  32. This is very interesting, You are a very skilled blogger. I ave joined your feed and look forward to seeking more of your fantastic post. Also, I have shared your site in my social networks!

  33. I really enjoy the blog.Really looking forward to read more. Keep writing.

  34. Delicious says:

    This blog was how do I say it? Relevant!! Finally I ave found something that helped me. Cheers!

  35. Photography says:

    A round of applause for your article post.Much thanks again. Really Great.

  36. Friv says:

    Wow, amazing blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your site is excellent, let alone the content!. Thanks For Your article about sex.

  37. events says:

    Wow, great blog post.Thanks Again. Want more.

  38. ItaаАа’б‚Т€ТšаЂаŒаАа’б‚Т€ТžаБТžs tremendous blog, I wish to be like you

  39. I truly appreciate this blog article.Thanks Again. Keep writing.

  40. Muchos Gracias for your blog. Keep writing.

  41. Im obliged for the blog post.Much thanks again. Great.

  42. I wanted to thank you for this great read!! I definitely enjoying every little bit of it I have you bookmarked to check out new stuff you post.

  43. Thank you ever so for you article post.Really looking forward to read more. Really Cool.

  44. Looking forward to reading more. Great article post.Thanks Again. Really Great.

  45. There is perceptibly a bundle to know about this. I suppose you made some nice points in features also.

  46. Some truly prize posts on this web site, saved to favorites.

  47. I was recommended this blog by my cousin. I am not sure whether this post is written by him as nobody else know such detailed about my difficulty. You are incredible! Thanks!

  48. Major thankies for the blog post.Much thanks again. Cool.

  49. pest control says:

    Im thankful for the blog article.Thanks Again. Really Great.

  50. Buzzinity says:

    Wow, incredible blog layout! How long have you

  51. Muchos Gracias for your article.Much thanks again. Really Cool.

  52. You ave made some good points there. I looked on the net for additional information about the issue and found most people will go along with your views on this website.

  53. Really informative article. Really Cool.

  54. Click Here says:

    I think this is a real great post.Much thanks again. Fantastic.

  55. I really like and appreciate your blog post.Really looking forward to read more. Keep writing.

  56. lumenhub says:

    You cann at imagine just how much time I had spent for this information! Thanks!

  57. Enjoyed every bit of your article.Thanks Again. Awesome.

  58. Bethanie says:

    S design houses In third place is michael kors canada, rising two spots in the ranks from Tuesday,

  59. Awesome blog. Really Great.

  60. IMDB visits says:

    I truly appreciate this article post.Really looking forward to read more. Great.

  61. Very neat post.Much thanks again.

  62. Entry-level says:

    I value the article.Thanks Again. Cool.

  63. Huong says:

    Im obliged for the blog.Really looking forward to read more. Really Cool.

  64. I value the article post.

  65. Artist says:

    Your style is unique compared to other people I have read stuff from. Thank you for posting when you ave got the opportunity, Guess I will just book mark this site.

  66. Utterly written articles, thanks for entropy.

  67. This website was how do I say it? Relevant!! Finally I have found something that helped me. Thank you!

  68. Nightclub says:

    Awesome article.Really looking forward to read more. Cool.

  69. Just wanna tell that this is extremely helpful, Thanks for taking your time to write this.

  70. #weedmaps says:

    Fantastic blog.Thanks Again. Awesome.

  71. This blog was how do I say it? Relevant!! Finally I have found something that helped me. Kudos!

  72. Very nice blog post. I certainly love this site. Continue the good work!

  73. Keep up the good work , I read few blog posts on this web site and I conceive that your web blog is very interesting and has got sets of excellent info.

  74. This very blog is definitely interesting and diverting. I have picked a lot of interesting advices out of it. I ad love to return over and over again. Cheers!

  75. I?аАТ’аЂа†ll right away take hold of your rss as I can at find your e-mail subscription link or e-newsletter service. Do you ave any? Please allow me recognize so that I could subscribe. Thanks.

  76. Hey, thanks for the post.Much thanks again. Awesome.

  77. That is a really good tip particularly to those fresh to the blogosphere. Simple but very precise informationaаАа’б‚Т€ТšаЂаŒаАТ’аЂа† Many thanks for sharing this one. A must read post!

  78. sex says:

    Thanks-a-mundo for the article.Thanks Again. Cool.

  79. i use google when i want to do some spanish translation, it is good for general spanish translation.,

  80. Wow, what a video it is! Really fastidious quality video, the lesson given in this video is actually informative.

  81. flexvit says:

    Im thankful for the blog post.Really thank you! Keep writing.

  82. out. I like what I see so now i am following you.

  83. eleaf pico says:

    This very blog is without a doubt awesome and besides factual. I have found a lot of handy tips out of this source. I ad love to come back every once in a while. Thanks a lot!

  84. Looking forward to reading more. Great article post.Much thanks again. Want more.

  85. Im thankful for the article post.Really thank you! Really Cool.

  86. Thank you ever so for you article. Will read on

  87. Wilbgraimi says:

    Acquistare viagra 25 mg [url=http://viag1.xyz/cheap-cheap-viagra.php]Cheap Cheap Viagra[/url] Order Flagyl In Sacramento What Is Cortanmycetine Used For [url=http://zol1.xyz/buying-zoloft-online.php]Buying Zoloft Online[/url] Kamagra Women Pfizer Viagra Without Prescription [url=http://cial1.xyz/purchase-cheap-cialis.php]Purchase Cheap Cialis[/url] Special Care Pharmacy Puerto Rico Hydrochlorothiazide Order Secure Ordering With Free Shipping Shop [url=http://zol1.xyz/zoloft-on-line.php]Zoloft On Line[/url] Usa No Prescription Generic Viagra Wat Is Kamagra [url=http://cial1.xyz/generic-cialis-usa.php]Generic Cialis Usa[/url] Purchase Cialis From North America Vente Cialis En France [url=http://cial1.xyz/order-cialis-online.php]Order Cialis Online[/url] Levitra 20 Mg Trucchi How Long Does Amoxicillin Stay Good [url=http://kama1.xyz/cheap-kamagra-tablets.php]Cheap Kamagra Tablets[/url] Zithromax Fungal Infection Where Can You Buy Arestin [url=http://viag1.xyz/cheap-viagra-fast.php]Cheap Viagra Fast[/url] Cialis Uso Cytotec For Sale [url=http://nolvadex.ccrpdc.com/nolvadex-price.php]Nolvadex Price[/url] Viagra Bas Prix Propecia Online Purchase [url=http://viag1.xyz/cheap-cheap-viagra.php]Cheap Cheap Viagra[/url] Keflex Dose Uti Emedicine Comprare Cialis Germania [url=http://cial1.xyz/cheap-cialis-tablets.php]Cheap Cialis Tablets[/url] Cephalexin Use’S

  88. WWW says:

    Very informative blog article. Really Great.

  89. Just Browsing While I was browsing today I noticed a great article about

  90. Im no pro, but I suppose you just made the best point. You certainly fully understand what youre talking about, and I can seriously get behind that. Thanks for being so upfront and so sincere.

  91. pretty handy stuff, overall I imagine this is really worth a bookmark, thanks

  92. Letitia says:

    Oakley has been gone for months, but the

  93. Wow! Thank you! I continually wanted to write on my blog something like that. Can I take a part of your post to my website?

  94. incredibly great submit, i really appreciate this internet internet site, carry on it

  95. Wow, what a video it is! Genuinely fastidious quality video, the lesson given in this video is truly informative.

  96. Taoism says:

    You made a few nice points there. I did a search on the subject and found most folks will have the same opinion with your blog.

  97. click here says:

    I really liked your article post.Really looking forward to read more. Cool.

  98. Thanks again for the blog article.Really thank you!

  99. Really appreciate you sharing this blog post.Much thanks again. Will read on

  100. Great blog. Really Great.

  101. App web says:

    Regards for helping out, wonderful info. аЂа‹аЂ Our individual lives cannot, generally, be works of art unless the social order is also.аЂ аЂа› by Charles Horton Cooley.

  102. Webdesign says:

    This is a excellent blog, would you be interested in doing an interview about just how you designed it? If so e-mail me!

  103. Very good blog article.Much thanks again. Want more.

  104. to go to see the web site, that as what this web page is providing.

  105. Wow, great blog post.Much thanks again. Really Great.

  106. What as Happening i am new to this, I stumbled upon this I ave found It absolutely useful and it has helped me out loads. I hope to contribute & assist other users like its helped me. Great job.

  107. Wow, great article.Really thank you! Want more.

  108. Well I sincerely liked studying it. This tip offered by you is very practical for correct planning.

  109. Very good article. Much obliged.

  110. Some really prime blog posts on this site, saved to favorites.

  111. backpacking says:

    I loved your blog.Much thanks again. Really Cool.

  112. Im obliged for the blog.Really thank you! Really Cool.

  113. Remarkable! Its in fact amazing article, I have got much clear idea on the topic of from this paragraph.

  114. The Silent Shard This may most likely be very handy for a few of your work opportunities I intend to you should not only with my blogging site but

  115. to be good. I have bookmarked it in my google bookmarks.

Leave a Reply

Your email address will not be published.