Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 360720 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0769 1209400 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0769 1209400 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1491 4102408 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1507 4130000 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1516 4147712 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1516 4164720 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1525 4172456 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1525 4172456 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2195 4408120 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2211 4425520 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2217 4446448 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2316 4514456 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2316 4514712 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360720 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0769 1209400 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0769 1209400 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1491 4102408 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1507 4130000 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1516 4147712 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1516 4164720 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1525 4172456 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1525 4172456 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2195 4408120 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2211 4425520 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2217 4446448 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3546 12305344 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3549 12305792 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3549 12305792 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3551 12314040 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3553 12315776 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3559 12348824 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3560 12349224 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3561 12349304 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

The Unique Factorisation Theorem

Every polynomial of degree n can be factorised intolinear factors, some of which may be repeated, so that ifis factorised asthen

The constantand the rootsmay be either real or complex. The factorisation is unique apart from the order of the factors. This means that the orderof each rootis well defined.

If we have two polynomials with the same factorisation apart from the order of the factors, then we can reorder the factors of one polynomial to match the order of the other. Expanding the brackets for both factorisations then returns identical polynomials, or cancelling brackets and powers, one at a time returns 1=1.

Every polynomialwhose coefficientsare all real can be expressed uniquely as a product of real linear and real quadratic factors. For any real quadratic factorwhich cannot be written as a product of real linear factors, the discriminant

Ifis a root of a polynomial with real coefficients then its complex conjugate is also a root and these roots exist for any polynomial whose factorisation into real factors contains the quadratic giving rise to the rootsand

Every polynomial of odd degree has at least one real zero, since we can factorise the polynomial either into three real linear factors, with each real linear factor returning one real root, or a real linear factor and a real quadratic, with the real linear factor giving one root and the quadratic returning complex conjugate roots.

Add comment

Security code
Refresh