ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Factoring Part 3

Updated on March 31, 2012

The case we did not cover in my two previous hubs on factoring is a trinomial with leading coefficient 1 in which the middle term is either positive or negative and the last term is negative. This is the most challenging of the three because there are more sign possibilities with the factors. This means that you cannot count on both numbers being positive or both being negative. In fact, one number must be positive and one must be negative. The question is WHO GETS THE POSITIVE AND WHO GETS THE NEGATIVE. Now that all sounds complicated but i can greatly simplify things for you by providing the following statement: the sign when adding a positive and a negative is the SIGN OF THE LARGER NUMBER.

Let me demonstrate. Suppose we want to factor x^2 + 3x - 18. The possible factorizations of 18 are 1*18, 2*9, and 3*6. One of these numbers must be positive and the other must be negative because we want them to multiply and produce a negative 18. But when we add them we would like positive 3 to be the result. According to what i said above THE LARGER NUMBER SHOULD BE POSITIVE. Therefore our possibilities are really -1*18, -2*9, and -3*6. The last pairing here will give us what we want. -3*6 = -18 and -3+6 = 3. We conclude that x^2 + 3x - 18 = (x-3)(x+6).

Let's try another one. How about x^2 - 4x - 5. In this particular case there are only two numbers which multiply and equal 5, those being 1 and 5. Here we want THE LARGER NUMBER TO BE NEGATIVE because we ultimately want -4 after adding them. So quickly we see that 1 and -5 will do the job and we arrive at the answer (x+1)(x-5). The two examples illustrate that in this situation if the middle number is positive we stick a positive on the larger number and if the middle number is negative we stick a negative on the larger number. Not too bad then is it? With this in mind the two trinomials x^2 - 1x - 30 and x^2 + 5x - 14 become quickly factorable. -6*5=-30 and -6+5=-1(larger number must be negative) are satisfactory equations for the first problem as are 7*-2=-14 and 7+-2=5(larger number must be positive) for the second problem. (x-6)(x+5) and (x+7)(x-2) are then the factored trinomials. Try x^2 + 4x - 96 and x^2 - 6x - 27 for practice.


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)