Login

FACTORISATION ET PRODUITS, IDENTITES REMARQUABLES

8ème Education de base - Algèbre

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

FACTORISATION ET PRODUITS, IDENTITES REMARQUABLES

Domaine	Science	Sous domaine	Mathématiques
Section	Cycle d'Orientation (C.O)	Option	Education de base
Discipline	Algèbre	Classe	8ème
Matériel didactique	Machines Scientifiques	Auteur	SCHOOLAP.COM
Objectif opérationnel	A la fin de la leçon, l'élève sera capable de maîtriser les procédés pouvant amener à des résolutions liées aux factorisations.
Réference	Bibliographie : KAYEMBE et Cie, Maîtriser les maths 2, Programme national de math, 2005, page 183.
Activité initiale
Rappel Divisez : 12 a² b³ - 20 a³ b⁴ + 16 a⁵ b² : 4 a b²		Rappel Divisez : 12 a² b³ - 20 a³ b⁴ + 16 a⁵ b² : 4 a b².
Motivation Comment s'appelle un procédé qui consiste à mettre une somme des termes en un produit de facteurs ?		Motivation Un procédé qui consiste à mettre une somme des termes en un produit de facteurs s'appelle la factorisation.
Annonce du Sujet Qu'allons-nous étudier aujourd'hui ?		Annonce du Sujet Aujourd'hui, nous allons étudier la factorisation et produits, identités remarquables.
Activité principale
Analyse Comment peut-on réaliser la factorisation ? Quand y a-t-il mise en évidence ?		Analyse Pour réaliser la factorisation en facteurs, on utilise les méthodes de décomposition telles que : La mise en évidence; L'emploi des identités remarquables; Le trinôme du second degré. La mise en évidence Il y a mise en évidence lorsque tous les termes d'une expression algébrique renferment un facteur commun. 20 a² x⁴ - 16 a² x² - 24 a³ x = 4 a² x (5 x³ - 4 x + 6 a) 132 x + 360 y - 84 = 12 (11 x + 30 y - 7) 15 a² x³ y⁴ + 5 a⁴ x⁵ y⁶ - 5 a x² y³ = 5 x² y³ (3 a x y + a³ x³ y³-1)
Comment peut-on employer les identités remarquables ? Différence de deux carrées (a-b) (a + b) = a² + ab - ab - b² Trinôme carré parfait ( a + b )² = a² + 2 a b + b² (a - b)² = ( 3 x + 2 )² = ( 3 x + 2 ) ( 3 x + 2 ) = 9 x² + 6 x + 6 x + 4 9 x² + 12 x + 4 Quadrinôme cube parfait ( a + b )³ = a³ + 3 a² b + 3 a b² + b³ (a - b)³ = a³- 3 a² b + 3 a b²- b³ (2 x - y²)³ = 8 x³ - 12 x² y² + 6 x y⁴ - y⁶ ( x + 1)³ = x³ + 3 x + 1 + 3 x³ Somme et différence de deux cubes a³ + b³ = (a + b) (a²- a b + b²) a³- b³ = (a - b) (a² + a b + b²) Trinôme du second degré Un trinôme du second degré en x est tout trinôme de la forme a x² + b x + c (1) ( a ǂ 0 )		Emploi des identités remarquables Différence de deux carrées (a-b) (a + b) = a² + ab - ab - b² Trinôme carré parfait ( a + b )² = a² + 2 a b + b² (a - b)² = ? ( 3 x + 2 )² = ( 3 x + 2 ) ( 3 x + 2 ) = 9 x² + 6 x + 6 x + 4 9 x² + 12 x + 4 Quadrinôme cube parfait ( a + b )³ = a³ + 3 a² b + 3 a b² + b³ (a - b)³ = a³- 3 a² b + 3 a b²- b³ (2 x - y²)³ = 8 x³ - 12 x² y² + 6 x y⁴ - y⁶ ( x + 1)³ = x³ + 3 x + 1 + 3 x³ Somme et différence de deux cubes a³ + b³ = (a + b) (a²- a b + b²) a³- b³ = (a - b) (a² + a b + b²) Trinôme du second degré Un trinôme du second degré en x est tout trinôme de la forme a x² + b x + c (1) ( a ǂ 0 )
Synthèse
Qu'est-ce que nous venons de voir ?		Nous venons de voir la factorisation et la mise en évidence.

Messages Timeline Exceptions Views7 Route Queries14 Models19 Gate Session Request

8.1.28310ms5MBGET fiches/{id}/{slug}

Booting (38.05ms)
Application (272ms)
1 x Application (87.71%)
272.16ms
1 x Booting (12.26%)
38.05ms

7 templates were rendered

public.fiches.show (resources/views/public/fiches/show.blade.php)2bladefile
Params
0
post
1
posts
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
post
4
posts
5
component
6
__currentLoopData
7
post_
8
loop
9
content
layouts.include.public.header (resources/views/layouts/include/public/header.blade.php)10bladefile
Params
0
__env
1
app
2
errors
3
post
4
posts
5
component
6
__currentLoopData
7
post_
8
loop
9
content
layouts.include.public.footer (resources/views/layouts/include/public/footer.blade.php)10bladefile
Params
0
__env
1
app
2
errors
3
post
4
posts
5
component
6
__currentLoopData
7
post_
8
loop
9
content

uri: GET fiches/{id}/{slug}
middleware: web
controller: App\Http\Controllers\Public\FicheController@show
as: public.fiche.show
namespace
prefix: /fiches
where
file: app/Http/Controllers/Public/FicheController.php:46-73

14 statements were executed239ms

select * from `oldt_posts` where `oldt_posts`.`id` = '2529' limit 1

2.48ms/app/Repositories/PostRepository.php:39schoolap

Metadata
Bindings	0. 2529
Backtrace	19. /app/Repositories/PostRepository.php:39 20. /app/Http/Controllers/Public/FicheController.php:48 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 (`post_id` = 2529) and `lessons`.`deleted_at` is null limit 1

700μs/app/Http/Controllers/Public/FicheController.php:52schoolap

Metadata
Bindings	0. 2529
Backtrace	15. /app/Http/Controllers/Public/FicheController.php:52 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

update `lessons` set `count_view` = `count_view` + 1, `lessons`.`updated_at` = '2025-05-08 06:18:48' where `id` = 2867 and `lessons`.`deleted_at` is null

4.19ms/app/Http/Controllers/Public/FicheController.php:55schoolap

Metadata
Bindings	0. 2025-05-08 06:18:48 1. 2867
Backtrace	16. /app/Http/Controllers/Public/FicheController.php:55 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 `oldt_posts` where `published` = 1 order by RAND() limit 4

65.28ms/app/Http/Controllers/Public/FicheController.php:61schoolap

Metadata
Bindings	0. 1
Backtrace	14. /app/Http/Controllers/Public/FicheController.php:61 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

select * from `oldt_material_sheets` where `oldt_material_sheets`.`id` = 2145 limit 1

1.03msview::public.fiches.show:7schoolap

Metadata
Bindings	0. 2145
Backtrace	20. view::public.fiches.show:7 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 `oldt_grades` where `oldt_grades`.`id` = 8 and `oldt_grades`.`id` is not null limit 1

570μsview::public.fiches.show:15schoolap

Metadata
Bindings	0. 8
Backtrace	20. view::public.fiches.show:15 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 `oldt_options` where `oldt_options`.`id` = 15 limit 1

570μsview::public.fiches.show:15schoolap

Metadata
Bindings	0. 15
Backtrace	20. view::public.fiches.show:15 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 `oldt_disciplines` where `oldt_disciplines`.`id` = 1 limit 1

610μsview::public.fiches.show:15schoolap

Metadata
Bindings	0. 1
Backtrace	20. view::public.fiches.show:15 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 `oldt_sub_domains` where `oldt_sub_domains`.`id` = 1 and `oldt_sub_domains`.`id` is not null limit 1

560μsview::public.fiches.show:120schoolap

Metadata
Bindings	0. 1
Backtrace	20. view::public.fiches.show:120 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 `oldt_domains` where `oldt_domains`.`id` = 1 and `oldt_domains`.`id` is not null limit 1

510μsview::public.fiches.show:120schoolap

Metadata
Bindings	0. 1
Backtrace	20. view::public.fiches.show:120 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 `oldt_sections` where `oldt_sections`.`id` = 7 limit 1

500μsview::public.fiches.show:136schoolap

Metadata
Bindings	0. 7
Backtrace	20. view::public.fiches.show:136 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 `oldt_material_sheet_contents` where `material_sheet_id` = 2145 and `activity` = 0

53.91ms/app/Models/Old/MaterialSheet.php:73schoolap

Metadata
Bindings	0. 2145 1. 0
Backtrace	14. /app/Models/Old/MaterialSheet.php:73 15. view::public.fiches.show:191 17. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 18. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 19. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70

select * from `oldt_material_sheet_contents` where `material_sheet_id` = 2145 and `activity` = 1

55.69ms/app/Models/Old/MaterialSheet.php:73schoolap

Metadata
Bindings	0. 2145 1. 1
Backtrace	14. /app/Models/Old/MaterialSheet.php:73 15. view::public.fiches.show:203 17. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 18. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 19. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70

select * from `oldt_material_sheet_contents` where `material_sheet_id` = 2145 and `activity` = 2

52.54ms/app/Models/Old/MaterialSheet.php:73schoolap

Metadata
Bindings	0. 2145 1. 2
Backtrace	14. /app/Models/Old/MaterialSheet.php:73 15. view::public.fiches.show:214 17. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 18. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 19. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70

App\Models\Old\MaterialSheetContent: 6
App\Models\Old\Section: 1
App\Models\Old\Domain: 1
App\Models\Old\SubDomain: 1
App\Models\Old\Discipline: 1
App\Models\Old\Option: 1
App\Models\Old\Grade: 1
App\Models\Old\MaterialSheet: 1
App\Models\Lesson\Lesson: 1
App\Models\Old\Post: 5

_token: thakrkwqgIKG0LVNdZvDyFrvCgEllF2JUUicMEu6
applocale: en
_previous: array:1 [ "url" => "https://s8.schoolap.com/fiches/2529/factorisation-et-produits-identites-remarq...
_flash: array:2 [ "old" => [] "new" => [] ]
PHPDEBUGBAR_STACK_DATA: []

path_info

/fiches/2529/factorisation-et-produits-identites-remarquables

status_code

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" => "54018"
  "REMOTE_ADDR" => "18.223.247.164"
  "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" => "/fiches/2529/factorisation-et-produits-identites-remarquables"
  "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" => 1746685127.96
  "REQUEST_TIME" => 1746685127
]

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 => "Thu, 08 May 2025 06:18:48 GMT"
  ]
  "set-cookie" => array:3 [▶
    0 => "lang_schoolap=eyJpdiI6ImVyQTBmT1E3ZkpSUW9kUC9nOXNya0E9PSIsInZhbHVlIjoiVVJ5dEM3UytCcnRGR3hock9EZUhQWURYakZGckUvRGJPejJ1bjhlRTJZbXVuM2kyVUtIeVVJMng4L2JYanJDWSIsIm1hYyI6Ijc4ZWFlZDBjYmQ1ZGYzZWRiYWQ1MDA0MGVmOGIzZjdkZDBiNTJlMWRiNTI5ZDZjNzc4ODA3NmUxYWVkZDcyZWYiLCJ0YWciOiIifQ%3D%3D; expires=Thu, 15 May 2025 04:58:48 GMT; Max-Age=600000; path=/; httponly; samesite=lax ◀lang_schoolap=eyJpdiI6ImVyQTBmT1E3ZkpSUW9kUC9nOXNya0E9PSIsInZhbHVlIjoiVVJ5dEM3UytCcnRGR3hock9EZUhQWURYakZGckUvRGJPejJ1bjhlRTJZbXVuM2kyVUtIeVVJMng4L2JYanJDWSIsIm ▶"
    1 => "XSRF-TOKEN=eyJpdiI6ImVTYnhiS1hMYUpoc2JaQkFJTXZwUXc9PSIsInZhbHVlIjoidGJZUDR3a3FadGlWWE5tY3phT1hDYTNhazNQSlhMd2NiMVlVRDZCZWc0VlovWUlESHhlVmRjamNJNXhFVUFTWGpmdlJ3R2MrbmZBaWN1MGJqSVJEdVNYRDRpMlFWQ1Q3MzNVdGxsWHVoZGhCUGFYdFB0S1NHQjlYdzdYZGIwWmQiLCJtYWMiOiIwNWE0YzZmZjBiZmFjOWJmOGU2N2QyOGQ4OGEzM2ZmYzZmNWU3NGU2MjZkMzkwNWVmNDU0OWQ2NmU2MTVlMTMzIiwidGFnIjoiIn0%3D; expires=Thu, 08 May 2025 08:18:48 GMT; Max-Age=7200; path=/; samesite=lax ◀XSRF-TOKEN=eyJpdiI6ImVTYnhiS1hMYUpoc2JaQkFJTXZwUXc9PSIsInZhbHVlIjoidGJZUDR3a3FadGlWWE5tY3phT1hDYTNhazNQSlhMd2NiMVlVRDZCZWc0VlovWUlESHhlVmRjamNJNXhFVUFTWGpmdlJ3R ▶"
    2 => "schoolap_session=eyJpdiI6InlDd3c1cFU0bzh2d3RYcDVNc2hDbHc9PSIsInZhbHVlIjoiWHYvRWoxNEUxdUt1dUh0OTFWVjYxbVhFTkNtdDFOdDk4NGsxUVUyem9VSms4SWZjWDdSSDN3SG82dTRhZXkyNzNoMWJjcERuM2FVSGJSVEp6a0gvNW1QRnd3KytEdXhxa2F0VnVxRVhVb1hXUnQ2SWk1ZnlqY1NTR3owclU1aFUiLCJtYWMiOiIxYTc0MjI4NjIyZWI1N2EwNTJlNGU1NWU3YzI4ZDlhMTU3NTA5Njc3Y2YzMGNjOWVhNTNhY2IyZTE1ZGVmYjhjIiwidGFnIjoiIn0%3D; expires=Thu, 08 May 2025 08:18:48 GMT; Max-Age=7200; path=/; httponly; samesite=lax ◀schoolap_session=eyJpdiI6InlDd3c1cFU0bzh2d3RYcDVNc2hDbHc9PSIsInZhbHVlIjoiWHYvRWoxNEUxdUt1dUh0OTFWVjYxbVhFTkNtdDFOdDk4NGsxUVUyem9VSms4SWZjWDdSSDN3SG82dTRhZXkyNzN ▶"
  ]
  "Set-Cookie" => array:3 [▶
    0 => "lang_schoolap=eyJpdiI6ImVyQTBmT1E3ZkpSUW9kUC9nOXNya0E9PSIsInZhbHVlIjoiVVJ5dEM3UytCcnRGR3hock9EZUhQWURYakZGckUvRGJPejJ1bjhlRTJZbXVuM2kyVUtIeVVJMng4L2JYanJDWSIsIm1hYyI6Ijc4ZWFlZDBjYmQ1ZGYzZWRiYWQ1MDA0MGVmOGIzZjdkZDBiNTJlMWRiNTI5ZDZjNzc4ODA3NmUxYWVkZDcyZWYiLCJ0YWciOiIifQ%3D%3D; expires=Thu, 15-May-2025 04:58:48 GMT; path=/; httponly ◀lang_schoolap=eyJpdiI6ImVyQTBmT1E3ZkpSUW9kUC9nOXNya0E9PSIsInZhbHVlIjoiVVJ5dEM3UytCcnRGR3hock9EZUhQWURYakZGckUvRGJPejJ1bjhlRTJZbXVuM2kyVUtIeVVJMng4L2JYanJDWSIsIm ▶"
    1 => "XSRF-TOKEN=eyJpdiI6ImVTYnhiS1hMYUpoc2JaQkFJTXZwUXc9PSIsInZhbHVlIjoidGJZUDR3a3FadGlWWE5tY3phT1hDYTNhazNQSlhMd2NiMVlVRDZCZWc0VlovWUlESHhlVmRjamNJNXhFVUFTWGpmdlJ3R2MrbmZBaWN1MGJqSVJEdVNYRDRpMlFWQ1Q3MzNVdGxsWHVoZGhCUGFYdFB0S1NHQjlYdzdYZGIwWmQiLCJtYWMiOiIwNWE0YzZmZjBiZmFjOWJmOGU2N2QyOGQ4OGEzM2ZmYzZmNWU3NGU2MjZkMzkwNWVmNDU0OWQ2NmU2MTVlMTMzIiwidGFnIjoiIn0%3D; expires=Thu, 08-May-2025 08:18:48 GMT; path=/ ◀XSRF-TOKEN=eyJpdiI6ImVTYnhiS1hMYUpoc2JaQkFJTXZwUXc9PSIsInZhbHVlIjoidGJZUDR3a3FadGlWWE5tY3phT1hDYTNhazNQSlhMd2NiMVlVRDZCZWc0VlovWUlESHhlVmRjamNJNXhFVUFTWGpmdlJ3R ▶"
    2 => "schoolap_session=eyJpdiI6InlDd3c1cFU0bzh2d3RYcDVNc2hDbHc9PSIsInZhbHVlIjoiWHYvRWoxNEUxdUt1dUh0OTFWVjYxbVhFTkNtdDFOdDk4NGsxUVUyem9VSms4SWZjWDdSSDN3SG82dTRhZXkyNzNoMWJjcERuM2FVSGJSVEp6a0gvNW1QRnd3KytEdXhxa2F0VnVxRVhVb1hXUnQ2SWk1ZnlqY1NTR3owclU1aFUiLCJtYWMiOiIxYTc0MjI4NjIyZWI1N2EwNTJlNGU1NWU3YzI4ZDlhMTU3NTA5Njc3Y2YzMGNjOWVhNTNhY2IyZTE1ZGVmYjhjIiwidGFnIjoiIn0%3D; expires=Thu, 08-May-2025 08:18:48 GMT; path=/; httponly ◀schoolap_session=eyJpdiI6InlDd3c1cFU0bzh2d3RYcDVNc2hDbHc9PSIsInZhbHVlIjoiWHYvRWoxNEUxdUt1dUh0OTFWVjYxbVhFTkNtdDFOdDk4NGsxUVUyem9VSms4SWZjWDdSSDN3SG82dTRhZXkyNzN ▶"
  ]
]

session_attributes

  0 of 0   
array:5 [▼
  "_token" => "thakrkwqgIKG0LVNdZvDyFrvCgEllF2JUUicMEu6"
  "applocale" => "en"
  "_previous" => array:1 [▶
    "url" => "https://s8.schoolap.com/fiches/2529/factorisation-et-produits-identites-remarquables"
  ]
  "_flash" => array:2 [▶
    "old" => []
    "new" => []
  ]
  "PHPDEBUGBAR_STACK_DATA" => []
]