Sign in
Login

Chers finalistes, préparez-vous pour le grand jour avec nos contenus !

Des items de toutes les options taillés sur mesure pour que vous prépariez mieux vos épreuves

Start learning
FRACTION RATIONNELLE

Rappel

Calculer :

a.  (t+2)2

b .  (2x2-3)2

Motivation

Nommer l'écriture suivante :

Annonce du Sujet

Aujourd'hui nous allons étudier la fraction rationnelle.

Analyse

Définir une fonction rationnelle.

Comment peut-on calculer le dom de la fraction rationnelle?

Comment peut-on simplifier des fractions rationnelles?

L'addition est-elle semblable à la soustraction?

Comment peut-on effectuer la multiplication?

Calculer la division

Synthèse

Qu'est-ce que nous venons de voir?

Calculer :

a.  (t+2)2 =  t2 + 4t +4

b .  (2x2-3)2 =  4x4 + x2 + 9

L'écriture suivante est une fraction.

Aujourd'hui nous allons étudier la fraction rationnelle.

Soit f et g deux applications polynômes dans R.

On appelle fonction rationnelle dans R, la fonction q(x) = f(x) / g(x)

Ce rapport s'appelle fraction rationnelle f(x) et g(x) sont les termes de la fraction.

Simplification des fractions rationnelles

Simplifier une fraction rationnelle, c'est diviser les 2 termes de la fraction par leur pgcd.

                    . 

Nous venons de voir la fraction rationnelle.

MessagesTimelineExceptionsViews7RouteQueries16Models18GateSessionRequest
8.1.28PHP Version105msRequest Duration5MBMemory UsageGET lessons/{id}/{slug}Route
    • Booting (32.85ms)
    • Application (72.43ms)
    • 1 x Application (68.74%)
      72.43ms
      1 x Booting (31.18%)
      32.85ms
      7 templates were rendered
      • public.lessons.show (resources/views/public/lessons/show.blade.php)2bladefile
        Params
        0
        lesson
        1
        lessons
      • components.banner-page (resources/views/components/banner-page.blade.php)7bladefile
        Params
        0
        property
        1
        firstTitle
        2
        secondTitle
        3
        src
        4
        attributes
        5
        slot
        6
        __laravel_slots
      • components.ad-card-lesson-long (resources/views/components/ad-card-lesson-long.blade.php)7bladefile
        Params
        0
        summary
        1
        title
        2
        actionLink
        3
        actionText
        4
        attributes
        5
        slot
        6
        __laravel_slots
      • components.button-link (resources/views/components/button-link.blade.php)6bladefile
        Params
        0
        class
        1
        title
        2
        href
        3
        attributes
        4
        slot
        5
        __laravel_slots
      • layouts.public (resources/views/layouts/public.blade.php)10bladefile
        Params
        0
        __env
        1
        app
        2
        errors
        3
        lesson
        4
        lessons
        5
        component
        6
        __currentLoopData
        7
        lesson_
        8
        loop
        9
        section
      • layouts.include.public.header (resources/views/layouts/include/public/header.blade.php)10bladefile
        Params
        0
        __env
        1
        app
        2
        errors
        3
        lesson
        4
        lessons
        5
        component
        6
        __currentLoopData
        7
        lesson_
        8
        loop
        9
        section
      • layouts.include.public.footer (resources/views/layouts/include/public/footer.blade.php)10bladefile
        Params
        0
        __env
        1
        app
        2
        errors
        3
        lesson
        4
        lessons
        5
        component
        6
        __currentLoopData
        7
        lesson_
        8
        loop
        9
        section
      uri
      GET lessons/{id}/{slug}
      middleware
      web
      controller
      App\Http\Controllers\Public\LessonController@show
      as
      public.lesson.show
      namespace
      prefix
      /lessons
      where
      file
      app/Http/Controllers/Public/LessonController.php:86-138
      16 statements were executed40.57ms
      schoolap
      • select * from `oldt_posts` where (`id` = '2814' and `slug` = 'fraction-rationnelle') and exists (select * from `oldt_material_sheets` where `oldt_posts`.`material_sheet_id` = `oldt_material_sheets`.`id`) limit 1
        2ms/app/Http/Controllers/Public/LessonController.php:92schoolap
        Metadata
        Bindings
        • 0. 2814
        • 1. fraction-rationnelle
        Backtrace
        • 15. /app/Http/Controllers/Public/LessonController.php:92
        • 16. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 17. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 18. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `lessons` where `lessons`.`id` = '2814' and `lessons`.`deleted_at` is null limit 1
        510μs/app/Repositories/LessonRepository.php:41schoolap
        Metadata
        Bindings
        • 0. 2814
        Backtrace
        • 19. /app/Repositories/LessonRepository.php:41
        • 20. /app/Http/Controllers/Public/LessonController.php:95
        • 21. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 22. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 23. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `lessons` where `lessons`.`deleted_at` is null order by RAND() limit 4
        27.28ms/app/Http/Controllers/Public/LessonController.php:96schoolap
        Metadata
        Backtrace
        • 14. /app/Http/Controllers/Public/LessonController.php:96
        • 15. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 16. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 17. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 18. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • update `lessons` set `count_view` = `count_view` + 1, `lessons`.`updated_at` = '2025-05-14 11:23:24' where `id` = 2814 and `lessons`.`deleted_at` is null
        3.66ms/app/Http/Controllers/Public/LessonController.php:98schoolap
        Metadata
        Bindings
        • 0. 2025-05-14 11:23:24
        • 1. 2814
        Backtrace
        • 16. /app/Http/Controllers/Public/LessonController.php:98
        • 17. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `chapters` where `chapters`.`id` = 641 and `chapters`.`deleted_at` is null limit 1
        940μsview::public.lessons.show:10schoolap
        Metadata
        Bindings
        • 0. 641
        Backtrace
        • 20. view::public.lessons.show:10
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `course_classes` where `course_classes`.`id` = 297 and `course_classes`.`deleted_at` is null limit 1
        580μsview::public.lessons.show:10schoolap
        Metadata
        Bindings
        • 0. 297
        Backtrace
        • 20. view::public.lessons.show:10
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `classes` where `classes`.`id` in (123) and `classes`.`deleted_at` is null
        460μsview::public.lessons.show:10schoolap
        Metadata
        Backtrace
        • 25. view::public.lessons.show:10
        • 27. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 28. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 29. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 30. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `cycle_levels` where `cycle_levels`.`id` in (70)
        780μsview::public.lessons.show:10schoolap
        Metadata
        Backtrace
        • 30. view::public.lessons.show:10
        • 32. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 35. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `cycles` where `cycles`.`id` in (11)
        540μsview::public.lessons.show:10schoolap
        Metadata
        Backtrace
        • 35. view::public.lessons.show:10
        • 37. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 38. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 39. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 40. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `levels` where `levels`.`id` in (28)
        390μsview::public.lessons.show:10schoolap
        Metadata
        Backtrace
        • 35. view::public.lessons.show:10
        • 37. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 38. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 39. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 40. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `courses` where `courses`.`id` in (106) and `courses`.`deleted_at` is null
        440μsview::public.lessons.show:10schoolap
        Metadata
        Backtrace
        • 25. view::public.lessons.show:10
        • 27. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 28. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 29. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 30. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `lesson_sections` where `lesson_sections`.`lesson_id` = 2814 and `lesson_sections`.`lesson_id` is not null
        790μsview::public.lessons.show:116schoolap
        Metadata
        Bindings
        • 0. 2814
        Backtrace
        • 19. view::public.lessons.show:116
        • 21. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 22. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 24. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `lesson_contents` where `lesson_contents`.`id` = 29335 and `lesson_contents`.`deleted_at` is null limit 1
        650μsview::public.lessons.show:117schoolap
        Metadata
        Bindings
        • 0. 29335
        Backtrace
        • 20. view::public.lessons.show:117
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `content_texts` where `content_texts`.`id` = 29118 and `content_texts`.`deleted_at` is null limit 1
        660μsview::public.lessons.show:133schoolap
        Metadata
        Bindings
        • 0. 29118
        Backtrace
        • 20. view::public.lessons.show:133
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `lesson_contents` where `lesson_contents`.`id` = 29337 and `lesson_contents`.`deleted_at` is null limit 1
        430μsview::public.lessons.show:117schoolap
        Metadata
        Bindings
        • 0. 29337
        Backtrace
        • 20. view::public.lessons.show:117
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      • select * from `content_texts` where `content_texts`.`id` = 29120 and `content_texts`.`deleted_at` is null limit 1
        460μsview::public.lessons.show:133schoolap
        Metadata
        Bindings
        • 0. 29120
        Backtrace
        • 20. view::public.lessons.show:133
        • 22. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 23. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
        • 25. /vendor/laravel/framework/src/Illuminate/View/View.php:195
      App\Models\ContentText
      2
      App\Models\Lesson\LessonContent
      2
      App\Models\Lesson\LessonSection
      2
      App\Models\Course\Course
      1
      App\Models\Level
      1
      App\Models\Cycle
      1
      App\Models\CycleLevel
      1
      App\Models\Classe
      1
      App\Models\Course\CourseClass
      1
      App\Models\Chapter
      1
      App\Models\Lesson\Lesson
      5
        _token
        VoMZxOg2Oa6UQ0QpFPBUzb9jgl8WMzFwk1tfJoMz
        applocale
        en
        _previous
        array:1 [ "url" => "https://s8.schoolap.com/lessons/2814/fraction-rationnelle" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /lessons/2814/fraction-rationnelle
        status_code
        200
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        request_request
        []
        request_headers
          0 of 0   
        array:17 [▼
  "accept-encoding" => array:1 [▶
    0 => "gzip, deflate, br, zstd"
  ]
  "sec-fetch-dest" => array:1 [▶
    0 => "document"
  ]
  "sec-fetch-user" => array:1 [▶
    0 => "?1"
  ]
  "sec-fetch-mode" => array:1 [▶
    0 => "navigate"
  ]
  "sec-fetch-site" => array:1 [▶
    0 => "none"
  ]
  "accept" => array:1 [▶
    0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7"
  ]
  "user-agent" => array:1 [▶
    0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
  ]
  "upgrade-insecure-requests" => array:1 [▶
    0 => "1"
  ]
  "sec-ch-ua-platform" => array:1 [▶
    0 => ""Windows""
  ]
  "sec-ch-ua-mobile" => array:1 [▶
    0 => "?0"
  ]
  "sec-ch-ua" => array:1 [▶
    0 => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99""
  ]
  "cache-control" => array:1 [▶
    0 => "no-cache"
  ]
  "pragma" => array:1 [▶
    0 => "no-cache"
  ]
  "connection" => array:1 [▶
    0 => "keep-alive"
  ]
  "host" => array:1 [▶
    0 => "s8.schoolap.com"
  ]
  "content-length" => array:1 [▶
    0 => ""
  ]
  "content-type" => array:1 [▶
    0 => ""
  ]
]
        request_server
          0 of 0   
        array:41 [▼
  "USER" => "www-data"
  "HOME" => "/var/www"
  "HTTP_ACCEPT_ENCODING" => "gzip, deflate, br, zstd"
  "HTTP_SEC_FETCH_DEST" => "document"
  "HTTP_SEC_FETCH_USER" => "?1"
  "HTTP_SEC_FETCH_MODE" => "navigate"
  "HTTP_SEC_FETCH_SITE" => "none"
  "HTTP_ACCEPT" => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7"
  "HTTP_USER_AGENT" => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
  "HTTP_UPGRADE_INSECURE_REQUESTS" => "1"
  "HTTP_SEC_CH_UA_PLATFORM" => ""Windows""
  "HTTP_SEC_CH_UA_MOBILE" => "?0"
  "HTTP_SEC_CH_UA" => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99""
  "HTTP_CACHE_CONTROL" => "no-cache"
  "HTTP_PRAGMA" => "no-cache"
  "HTTP_CONNECTION" => "keep-alive"
  "HTTP_HOST" => "s8.schoolap.com"
  "REDIRECT_STATUS" => "200"
  "SERVER_NAME" => "s8.schoolap.com"
  "SERVER_PORT" => "443"
  "SERVER_ADDR" => "188.166.147.66"
  "REMOTE_PORT" => "55678"
  "REMOTE_ADDR" => "18.217.17.116"
  "SERVER_SOFTWARE" => "nginx/1.18.0"
  "GATEWAY_INTERFACE" => "CGI/1.1"
  "HTTPS" => "on"
  "REQUEST_SCHEME" => "https"
  "SERVER_PROTOCOL" => "HTTP/1.1"
  "DOCUMENT_ROOT" => "/var/www/s8.schoolap.com/public"
  "DOCUMENT_URI" => "/index.php"
  "REQUEST_URI" => "/lessons/2814/fraction-rationnelle"
  "SCRIPT_NAME" => "/index.php"
  "CONTENT_LENGTH" => ""
  "CONTENT_TYPE" => ""
  "REQUEST_METHOD" => "GET"
  "QUERY_STRING" => ""
  "SCRIPT_FILENAME" => "/var/www/s8.schoolap.com/public/index.php"
  "FCGI_ROLE" => "RESPONDER"
  "PHP_SELF" => "/index.php"
  "REQUEST_TIME_FLOAT" => 1747221804.6095
  "REQUEST_TIME" => 1747221804
]
        request_cookies
        []
        response_headers
          0 of 0   
        array:5 [▼
  "content-type" => array:1 [▶
    0 => "text/html; charset=UTF-8"
  ]
  "cache-control" => array:1 [▶
    0 => "no-cache, private"
  ]
  "date" => array:1 [▶
    0 => "Wed, 14 May 2025 11:23:24 GMT"
  ]
  "set-cookie" => array:3 [▶
    0 => "lang_schoolap=eyJpdiI6IkFzcU1jOEUwTEVKNVRYempCOForRkE9PSIsInZhbHVlIjoiSlh5Sk5RSExPTXYvb3ZVOStsaFBmeFNkNlZ6blpBWnRJeDBiVWNSTHdLRGF0WEVWTDNWRHIyY1lyZkt2ZjdVcCIsIm1hYyI6IjYwNTU4ZWI5NjE5NjE4NDI2MjhkYTZhZmM3MDhlYmU2MWFmM2M4MWFkMjNkOTY5ZWE1ZTA0NTdkZjY0MjRjYzkiLCJ0YWciOiIifQ%3D%3D; expires=Wed, 21 May 2025 10:03:24 GMT; Max-Age=600000; path=/; httponly; samesite=lax ◀lang_schoolap=eyJpdiI6IkFzcU1jOEUwTEVKNVRYempCOForRkE9PSIsInZhbHVlIjoiSlh5Sk5RSExPTXYvb3ZVOStsaFBmeFNkNlZ6blpBWnRJeDBiVWNSTHdLRGF0WEVWTDNWRHIyY1lyZkt2ZjdVcCIsIm ▶"
    1 => "XSRF-TOKEN=eyJpdiI6Ikpvd2VIVk94dGtUVjZYR0RzZWprZ2c9PSIsInZhbHVlIjoiSFZ2RmdrQzFSNXNxWlRYRnNKYUlZRnFORVY1aFk5cnpVRzk0N3FUSExsZHNuUTRBT0dhUXJ6YTlzOGViT01Ca0NYaGVySURkcGZtODRXUlltUnh3WEhMa1ZmRG1DSzVickRxRGtFTzQyLzJFd2h0OFZhSFlLTlRKd0ltTzVDeGgiLCJtYWMiOiI2NjA1YzYxZTRiMDc0YTkwZmM2YTMyODA4ZWVkMWM5N2JiYzljZjQ3ZTkwYzUxNWI1ZWFhMTIyZWUwMTAzODQ1IiwidGFnIjoiIn0%3D; expires=Wed, 14 May 2025 13:23:24 GMT; Max-Age=7200; path=/; samesite=lax ◀XSRF-TOKEN=eyJpdiI6Ikpvd2VIVk94dGtUVjZYR0RzZWprZ2c9PSIsInZhbHVlIjoiSFZ2RmdrQzFSNXNxWlRYRnNKYUlZRnFORVY1aFk5cnpVRzk0N3FUSExsZHNuUTRBT0dhUXJ6YTlzOGViT01Ca0NYaGVyS ▶"
    2 => "schoolap_session=eyJpdiI6IncrWFNPejUvTHFEMHpwWDVkc3FUZHc9PSIsInZhbHVlIjoieFNZY2FWYlRzOXRjNnRsZCtRQ3J4emk1WXZiY24weFhQelFYQlIxSjFxVlgvbG5JSE1NYTNRK2JlVE9ueWlpb3hveUhFS1RNYmMzbHcrbWlmSURFb1Y4cUdWQlNHRlpCY1J0Z3ZtZjZrcXNSN0VYTy9oSUxDT0RlTnlUVXhLN0MiLCJtYWMiOiJlNDRiY2IwNGJiMmM2YjA5ZjQyOWQxZGJiNDJlYmM2ZjI2N2VkYjE5YzgzMWNmNDM2MzUxNWQ4NWM5NTNjYzA3IiwidGFnIjoiIn0%3D; expires=Wed, 14 May 2025 13:23:24 GMT; Max-Age=7200; path=/; httponly; samesite=lax ◀schoolap_session=eyJpdiI6IncrWFNPejUvTHFEMHpwWDVkc3FUZHc9PSIsInZhbHVlIjoieFNZY2FWYlRzOXRjNnRsZCtRQ3J4emk1WXZiY24weFhQelFYQlIxSjFxVlgvbG5JSE1NYTNRK2JlVE9ueWlpb3h ▶"
  ]
  "Set-Cookie" => array:3 [▶
    0 => "lang_schoolap=eyJpdiI6IkFzcU1jOEUwTEVKNVRYempCOForRkE9PSIsInZhbHVlIjoiSlh5Sk5RSExPTXYvb3ZVOStsaFBmeFNkNlZ6blpBWnRJeDBiVWNSTHdLRGF0WEVWTDNWRHIyY1lyZkt2ZjdVcCIsIm1hYyI6IjYwNTU4ZWI5NjE5NjE4NDI2MjhkYTZhZmM3MDhlYmU2MWFmM2M4MWFkMjNkOTY5ZWE1ZTA0NTdkZjY0MjRjYzkiLCJ0YWciOiIifQ%3D%3D; expires=Wed, 21-May-2025 10:03:24 GMT; path=/; httponly ◀lang_schoolap=eyJpdiI6IkFzcU1jOEUwTEVKNVRYempCOForRkE9PSIsInZhbHVlIjoiSlh5Sk5RSExPTXYvb3ZVOStsaFBmeFNkNlZ6blpBWnRJeDBiVWNSTHdLRGF0WEVWTDNWRHIyY1lyZkt2ZjdVcCIsIm ▶"
    1 => "XSRF-TOKEN=eyJpdiI6Ikpvd2VIVk94dGtUVjZYR0RzZWprZ2c9PSIsInZhbHVlIjoiSFZ2RmdrQzFSNXNxWlRYRnNKYUlZRnFORVY1aFk5cnpVRzk0N3FUSExsZHNuUTRBT0dhUXJ6YTlzOGViT01Ca0NYaGVySURkcGZtODRXUlltUnh3WEhMa1ZmRG1DSzVickRxRGtFTzQyLzJFd2h0OFZhSFlLTlRKd0ltTzVDeGgiLCJtYWMiOiI2NjA1YzYxZTRiMDc0YTkwZmM2YTMyODA4ZWVkMWM5N2JiYzljZjQ3ZTkwYzUxNWI1ZWFhMTIyZWUwMTAzODQ1IiwidGFnIjoiIn0%3D; expires=Wed, 14-May-2025 13:23:24 GMT; path=/ ◀XSRF-TOKEN=eyJpdiI6Ikpvd2VIVk94dGtUVjZYR0RzZWprZ2c9PSIsInZhbHVlIjoiSFZ2RmdrQzFSNXNxWlRYRnNKYUlZRnFORVY1aFk5cnpVRzk0N3FUSExsZHNuUTRBT0dhUXJ6YTlzOGViT01Ca0NYaGVyS ▶"
    2 => "schoolap_session=eyJpdiI6IncrWFNPejUvTHFEMHpwWDVkc3FUZHc9PSIsInZhbHVlIjoieFNZY2FWYlRzOXRjNnRsZCtRQ3J4emk1WXZiY24weFhQelFYQlIxSjFxVlgvbG5JSE1NYTNRK2JlVE9ueWlpb3hveUhFS1RNYmMzbHcrbWlmSURFb1Y4cUdWQlNHRlpCY1J0Z3ZtZjZrcXNSN0VYTy9oSUxDT0RlTnlUVXhLN0MiLCJtYWMiOiJlNDRiY2IwNGJiMmM2YjA5ZjQyOWQxZGJiNDJlYmM2ZjI2N2VkYjE5YzgzMWNmNDM2MzUxNWQ4NWM5NTNjYzA3IiwidGFnIjoiIn0%3D; expires=Wed, 14-May-2025 13:23:24 GMT; path=/; httponly ◀schoolap_session=eyJpdiI6IncrWFNPejUvTHFEMHpwWDVkc3FUZHc9PSIsInZhbHVlIjoieFNZY2FWYlRzOXRjNnRsZCtRQ3J4emk1WXZiY24weFhQelFYQlIxSjFxVlgvbG5JSE1NYTNRK2JlVE9ueWlpb3h ▶"
  ]
]
        session_attributes
          0 of 0   
        array:5 [▼
  "_token" => "VoMZxOg2Oa6UQ0QpFPBUzb9jgl8WMzFwk1tfJoMz"
  "applocale" => "en"
  "_previous" => array:1 [▶
    "url" => "https://s8.schoolap.com/lessons/2814/fraction-rationnelle"
  ]
  "_flash" => array:2 [▶
    "old" => []
    "new" => []
  ]
  "PHPDEBUGBAR_STACK_DATA" => []
]