{"id":1701,"date":"2009-09-18T15:58:28","date_gmt":"2009-09-18T14:58:28","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=1701"},"modified":"2009-09-18T15:58:28","modified_gmt":"2009-09-18T14:58:28","slug":"the-unreasonable-ineffectiveness-of-factoring","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=1701","title":{"rendered":"The unreasonable ineffectiveness of factoring"},"content":{"rendered":"

In his blog post, Factoring Schmactoring<\/a>,\u00a0 Sam Shah took a look at how many quadratic equations of the form x^2+bx+c<\/em> = 0 (where b<\/em> and c<\/em> are integers) could be factored over the integers and produced a chart that was a powerful reminder of just how few of these there are.<\/p>\n

\"Sam's<\/a><\/p>\n

After reading his post I fired up Mathematica and set to work extending Sam’s original chart to include negative values of b<\/em> and c<\/em> – the result of which is shown below.\u00a0 Out of 441 different quadratic equations, only 76 have integer solutions which is only 17.23%, and yet these are the only ones that succumb to the heavily drilled and killed method of factoring.<\/p>\n

\"My<\/p>\n

Sam and his commentators discussed the serious question of whether or not we should even be teaching factoring these days (for the record, I think we should) but my concerns were much less lofty.\u00a0 I wondered if I’d make something pretty if I did a similar chart to the one above but for values of b<\/em> and c<\/em> ranging from -100 to 100.<\/p>\n

\"Pretty<\/p>\n

OK, so it’s not so pretty but it is possibly even more striking than the original charts – there is a LOT of red.\u00a0 Next, I wondered about values of a<\/em> (the x^2<\/em> coefficient) other than 1 and thought that this was the perfect situation for a Manipulate (Click on the image to download an interactive version which can be used in either Mathematica or the free Mathematica Player<\/a>):<\/p>\n

\"Factoring<\/a><\/p>\n

Of course, when dealing with values of a<\/em> other than one, we should include the possibility of rational solutions so let’s do that and assign rational solutions to the colour blue – again click on the image below for the Mathematica-Payer compatible interactive version.<\/p>\n

\"Factoring<\/a><\/p>\n

I’m not really going anywhere with this – just having a bit of fun with Mathematica but comments are welcome.<\/p>\n

\n

\n","protected":false},"excerpt":{"rendered":"

In his blog post, Factoring Schmactoring,\u00a0 Sam Shah took a look at how many quadratic equations of the form x^2+bx+c = 0 (where b and c are integers) could be factored over the integers and produced a chart that was a powerful reminder of just how few of these there are. After reading his post […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6,8],"tags":[],"class_list":["post-1701","post","type-post","status-publish","format-standard","hentry","category-general-math","category-mathematica"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-rr","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1701","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1701"}],"version-history":[{"count":12,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1701\/revisions"}],"predecessor-version":[{"id":1713,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/1701\/revisions\/1713"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}