{"id":1158,"date":"2017-04-07T16:15:43","date_gmt":"2017-04-07T14:15:43","guid":{"rendered":"http:\/\/www2.mathnique.com\/site\/?page_id=1158"},"modified":"2018-06-09T00:33:20","modified_gmt":"2018-06-08T22:33:20","slug":"methodes","status":"publish","type":"page","link":"https:\/\/www.mathnique.com\/site\/methodes\/","title":{"rendered":"M\u00e9thodes et Outils"},"content":{"rendered":"<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"color: #ff0000;\"><b>R\u00e9sultats de base \u00e0 conna\u00eetre par coeur\u00a0<\/b><\/span>\n<ul>\n<li><span style=\"color: #ff0000;\"><strong>8 identit\u00e9s remarquables dans l'ensemble des r\u00e9els(mieux dans tout anneau commutatif) :<\/strong><\/span><\/li>\n<li><strong>$IR_1$ : $(a +b)^2 = a^2 + 2ab + b^2$<\/strong><br \/>\nd\u00e9monstration :<br \/>\n$(a +b)^2 =(a + b)(a + b)\u00a0= a^2 \u00a0+ab + ab + b^2 =a^2 + 2ab + b^2$<br \/>\nLe carr\u00e9 d'une somme est la somme du double produit et des carr\u00e9s des deux nombres.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1178 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/tetemort.png\" alt=\"\" width=\"103\" height=\"92\" \/>Attention, le carr\u00e9 d'une somme n'est pas la somme des carr\u00e9s d'autant plus que la fonction carr\u00e9 n'est pas une fonction lin\u00e9aire !<br \/>\nAttention, il ne faut jamais oublier le double produit : vous pouvez le v\u00e9rifier en observant bien\u00a0le sch\u00e9ma suivant :<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1512 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1-300x137.png\" alt=\"\" width=\"300\" height=\"137\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1-300x137.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1-768x351.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1-1024x468.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1-1200x548.png 1200w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/idr1.png 1222w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<li><b><\/b><strong>$IR_3$ : $(a -\u00a0b)^2 = a^2 -\u00a02ab + b^2$<\/strong><br \/>\nd\u00e9monstration :<br \/>\n- soit on remplace $b$ par $-b$ dans l'identit\u00e9 remarquable $IR_1$<br \/>\n- soit on calcule :\u00a0$(a -b)^2 =(a -\u00a0b)(a -\u00a0b)\u00a0= a^2 \u00a0-ab -\u00a0ab + b^2 =a^2 -\u00a02ab + b^2$<br \/>\n-\u00a0vous pouvez le v\u00e9rifier en observant bien\u00a0\u00a0le sch\u00e9ma suivant<sup> :<\/sup><sup>\u00a0<\/sup><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1517 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir2-300x140.png\" alt=\"\" width=\"300\" height=\"140\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir2-300x140.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir2-768x358.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir2-1024x478.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir2.png 1162w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<li><b><\/b><strong>$IR_3 $ :$(a + b)(a - b) =a^2 - b^2$<\/strong><br \/>\nD\u00e9monstration :<br \/>\n- par le calcul : $(a + b)(a - b) = a^2 -ab + ba - b^2 = a^2 - b^2$<br \/>\n- vous pouvez le v\u00e9rifier en observant bien\u00a0\u00a0le sch\u00e9ma suivant<sup> :<\/sup><sup><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1518 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir3-300x208.png\" alt=\"\" width=\"300\" height=\"208\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir3-300x208.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/ir3.png 748w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><br \/>\n<\/sup><\/li>\n<li><strong>$IR_4$ : $(a + b)^3 =a^3 + 3a^2 + 3ab^2 + b^3$<\/strong><\/li>\n<li><strong>$IR_5$ :$(a -\u00a0b)^3 =a^3 -\u00a03a^2 + 3ab^2 - b^3$<\/strong><span style=\"color: #ff0000;\"><strong><br \/>\n<\/strong><\/span>Il est assez facile de retenir ces identit\u00e9s remarquables en utilisant le triangle dit de PASCAL ou le triangle de SHU SHI YIE et la formule du bin\u00f4me de Newton :<\/li>\n<li><strong>$IR_6$ :\u00a0<\/strong><strong>$\\displaystyle{(a + b)^n = \\sum_{k = 0}^n \\binomial(n,k) a^k b^{n - k}}$<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-609 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-300x186.png\" alt=\"\" width=\"300\" height=\"186\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-300x186.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-768x477.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal-1024x636.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/trianglepascal.png 1178w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li><strong>$IR_7$ :\u00a0<\/strong>$a^3 + b^3 = ( a + b)(a^2 -ab + b^2)$<\/li>\n<li><strong>$IR_8$ :\u00a0<\/strong>$a^3 -\u00a0b^3 = ( a\u00a0-\u00a0b)(a^2 +ab + b^2$<\/li>\n<li><strong>$IR_9$ :\u00a0<\/strong>$(a + b + c)^2 =a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$<\/li>\n<li><b><strong>$IR_{10}$:\u00a0<\/strong><\/b><b>Pour tout entier $n$ &gt; 1,<br \/>\n$a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + \\cdots + ab^{n - 2} + b^{n - 1})$<\/b><\/p>\n<ul>\n<li><b>Exercice 1<br \/>\n<\/b>Calculer mentalement $101^2$\u00a0; $204^2$\u00a0; $101 \\times 99$ ; $394 \\times \u00a0406$ ; $47^2 - 43^2<span style=\"font-size: 12px;\">$<\/span><\/li>\n<li><b>Exercice 2<br \/>\n<\/b>Compl\u00e9ter les \u00e9galit\u00e9s suivantes :<br \/>\n$x^2 + \\cdots + 25 = \u00a0( ...... + ......)^2$<br \/>\n$x^2 + \\cdots + 16 = ( ...... + ......)^2$<br \/>\n$x^2 + \\cdots + 16 a^2\u00a0= ( ...... + ......)^2$<br \/>\n$x^2 + \\cdots + 16 a^2 = ( ...... -\u00a0......)^2$<br \/>\n$x^2 + 8x + \\cdots\u00a0\u00a0= ( ...... + ......)^2$<br \/>\n$x^2 - 10 x + \\cdots.= ( ...... -\u00a0......)^2$<br \/>\n$x^2 + 6x + \\cdots\u00a0= ( ...... + ......)^2$<br \/>\n$x^2 - 12 x +\\cdots\u00a0\u00a0= ( ...... -\u00a0......)^2$<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li><strong><span style=\"color: #ff0000;\">Th\u00e9or\u00e8me du signe du bin\u00f4me $ax +b$ avec $a \\neq 0$<br \/>\n<\/span><\/strong>\n<table id=\"tablepress-3\" class=\"tablepress tablepress-id-3\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">$x$<\/th><th class=\"column-2\">$-\\infty$<\/th><td class=\"column-3\"><\/td><th class=\"column-4\">            $\\frac{-b}{a}$<\/th><td class=\"column-5\"><\/td><th class=\"column-6\">$+\\infty$<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">$ax + b$<\/td><td class=\"column-2\"><\/td><td class=\"column-3\">signe contraire de $a$<\/td><td class=\"column-4\">                      0<\/td><td class=\"column-5\">signe de $a$<\/td><td class=\"column-6\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-3 from cache --><br \/>\nCe th\u00e9or\u00e8me permet de r\u00e9soudre ais\u00e9ment<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>une \u00e9quation du type $ax + b = 0$<\/li>\n<li>une in\u00e9quation du type $ax + b \\geq 0$<\/li>\n<li>une in\u00e9quation-produit du type $(ax + b)(cx + d) \\geq\u00a00$<\/li>\n<li>une in\u00e9quation-quotient du type $\\dfrac{ax + b}{cx +d} &lt; 0$<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"color: #ff0000;\"><strong>Th\u00e9or\u00e8me du signe du trin\u00f4me $ax^2 + bx + c$ avec $a \\neq 0$<\/strong><\/span>\n<ul>\n<li><strong><span style=\"color: #ff0000;\">1er cas : le\u00a0discriminant $\\Delta &lt; 0$<br \/>\n<\/span><\/strong><span style=\"color: #000000;\">\n<table id=\"tablepress-8\" class=\"tablepress tablepress-id-8\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">$x$<\/th><th class=\"column-2\">$-\\infty$<\/th><td class=\"column-3\"><\/td><th class=\"column-4\">$+\\infty$<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">$ax^2 + bx + c$<\/td><td class=\"column-2\"><\/td><td class=\"column-3\">           signe de $a$<\/td><td class=\"column-4\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-8 from cache --><\/span><strong><span style=\"color: #ff0000;\"><br \/>\n<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">2\u00e8me cas : le discriminant\u00a0<\/span><span style=\"color: #ff0000;\">$\\Delta &gt;\u00a00$<br \/>\n<\/span><\/strong>\n<table id=\"tablepress-9\" class=\"tablepress tablepress-id-9\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">$x$<\/th><th class=\"column-2\">$-\\infty$<\/th><td class=\"column-3\"><\/td><th class=\"column-4\">            $\\frac{-b}{2a}$<\/th><td class=\"column-5\"><\/td><th class=\"column-6\">$+\\infty$<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">$ax^2 + bx + c$<\/td><td class=\"column-2\"><\/td><td class=\"column-3\">     signe de $a$<\/td><td class=\"column-4\">                      0<\/td><td class=\"column-5\">      signe de $a$<\/td><td class=\"column-6\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-9 from cache --><strong><span style=\"color: #ff0000;\"><br \/>\n<\/span><\/strong><\/li>\n<li><strong><span style=\"color: #ff0000;\">3\u00e8me cas :\u00a0le discriminant\u00a0$\\Delta &gt;\u00a00$<br \/>\n<\/span><\/strong>\n<table id=\"tablepress-10\" class=\"tablepress tablepress-id-10\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">$x$<\/th><th class=\"column-2\">$-\\infty$<\/th><td class=\"column-3\"><\/td><th class=\"column-4\">              $x'$<\/th><td class=\"column-5\"><\/td><th class=\"column-6\">              $x\"$<\/th><td class=\"column-7\"><\/td><th class=\"column-8\">$+\\infty$<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">$ax^2 + bx + c$<\/td><td class=\"column-2\"><\/td><td class=\"column-3\">       signe de $a$<\/td><td class=\"column-4\">                 0<\/td><td class=\"column-5\"> signe contraire de   $a$<\/td><td class=\"column-6\">               0<\/td><td class=\"column-7\">          signe de $a$<\/td><td class=\"column-8\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-10 from cache --><strong><br \/>\n<\/strong>Ce th\u00e9or\u00e8me sous forme de 3 tableaux permet de r\u00e9soudre ais\u00e9ment<\/li>\n<li>une \u00e9quation du type $ax^2 + bx + c = 0$<\/li>\n<li>une in\u00e9quation du type $ax^2 + bx + c \\geq 0$<\/li>\n<li>une in\u00e9quation-produit du type $(ax^2 + bx + c)(dx^2 + ex + f) \\leq\u00a00$<\/li>\n<li>une in\u00e9quation-quotient du type $\\dfrac{ax^2 + bx + c}{dx^2 + ex + f} &lt; 0$<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"color: #ff0000;\"><strong>Comment r\u00e9soudre une \u00e9quation ou une in\u00e9quation d'inconnue $x$?<\/strong><\/span>\n<ul>\n<li>1\u00e8re \u00e9tape : chercher l'ensemble de d\u00e9finition $Def$ de cette \u00e9quation c'est-\u00e0-dire l'ensemble des $x$ pour lesquels les membres de cette \u00e9quation (ou in\u00e9quation) existent.<\/li>\n<li>2\u00e8me \u00e9tape : Pour tout $x \\in Def$ r\u00e9soudre cette \u00e9quation par \u00e9quivalence logique.<\/li>\n<li>3\u00e8me \u00e9tape : V\u00e9rifier que les solutions trouv\u00e9es appartient bien \u00e0 l'ensemble de d\u00e9finition puis \u00e9crire l'ensemble des solutions $\\mathcal{S}$<br \/>\n<span style=\"color: #008080;\"><em>Exemple :<\/em><\/span> R\u00e9soudre l'\u00e9quation suivante $\\dfrac{3x - 5}{x + 1} \u00a0 = \\dfrac{-7}{x} + \\dfrac{7 - x}{x^2 + x}$ d'inconnue $x \\in R$<\/p>\n<ul>\n<li>$Def =\\{ x \/ x + 1 \\neq 0 \\text{ et } x \\neq 0 \\text{ et } x^2 + x \\neq 0\\}$<br \/>\nComme $x + 1 = 0 \\iff x = -1$<br \/>\nComme $x^2 + x = 0 \\iff x(x + 1) = 0 \\iff x = 0 \\text{ ou } x = -1$<br \/>\nalors $Def = R - \\{ -1 ; 0\\}$<\/li>\n<li>$\\forall x \\in Def$ on a\u00a0$\\dfrac{3x - 5}{x + 1} \u00a0 = \\dfrac{-7}{x} + \\dfrac{7 - x}{x^2 + x}<br \/>\n\\iff \\dfrac{x(3x - 5)}{x(x + 1)} \u00a0 = \\dfrac{-7(x + 1)}{x(x + 1)} + \\dfrac{7 - x}{x^2 + x}$<br \/>\n$\\iff x(3x - 5) = -7(x + 1) + 7 - x \\iff 3x^3 - 5 x = -7 x - 7 + 7 - x \\iff 3x^2 -5x = -8x$<br \/>\n$\\iff 3x^2 + 3x = 0 \\iff 3x( x + 1) = \\iff x =0 \\text{ ou } x = -1$<\/li>\n<li>$0 \\notin Def$ et $-1 \\notin Def$ donc $\\mathcal{S} = \\emptyset$<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"color: #ff0000;\"><strong>Comment encadrer avec 13 r\u00e8gles de base ?<\/strong><\/span><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-521 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/genie.gif\" alt=\"\" width=\"128\" height=\"128\" \/><\/p>\n<ul>\n<li>$R_1$ : $a,m,m',M$ \u00e9tant\u00a0des \u00a0nombres r\u00e9els<br \/>\nSi $m \\leq a \\leq M$ alors $m + m' \\leq a \u00a0+ m' \\leq M + m'$<br \/>\nOn peut ajouter ou retrancher \u00a0aux 3 membres d'une double in\u00e9galit\u00e9 la m\u00eame valeur<\/li>\n<li>$R_2$ :<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1178 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/tetemort.png\" alt=\"\" width=\"103\" height=\"92\" \/><br \/>\nAttention , pour la multiplication de deux membres d'une in\u00e9galit\u00e9 par un m\u00eame nombre, il faut s'inqui\u00e9ter du signe de ce nombre.<br \/>\n$a,m,m',M$ \u00e9tant\u00a0des \u00a0nombres r\u00e9els<\/p>\n<ul>\n<li>Si $m' &gt;\u00a00$, Si $m \\leq a \\leq M$ alors $m m' \\leq a \u00a0m' \\leq M \u00a0m'$ : l'ordre est conserv\u00e9.<\/li>\n<li>Si $m' &lt;\u00a00$, Si $m \\leq a \\leq M$ alors $m m' \\geq a \u00a0m' \\geq M \u00a0m'$ : l'ordre est renserv\u00e9.<br \/>\nExemple classique : $ln(0.2) x \\leq ln(8) \\iff x \\geq \\dfrac{ln(3)}{ln(0,2)}$ car $ln(0,2) &lt; 0$ puisque $0&lt; ln(2) &lt; 1$<\/li>\n<\/ul>\n<\/li>\n<li>$R_3$ :\u00a0$a,b, m,m',M,M'$ \u00e9tant\u00a0des \u00a0nombres r\u00e9els<br \/>\nSi $m \\leq a \\leq M$<br \/>\nSi $m' \\leq b\u00a0\\leq M'$<br \/>\nalors\u00a0$m + m' \\leq a + b \\leq M + M'$<\/li>\n<li>$R_4$ :\u00a0$a,b, m,m',M,M'$ \u00e9tant\u00a0des \u00a0nombres r\u00e9els<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1178 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/tetemort.png\" alt=\"\" width=\"103\" height=\"92\" \/><br \/>\nSi $m \\leq a \\leq M$<br \/>\nSi $m' \\leq\u00a0b\u00a0\\leq M'$<br \/>\nalors\u00a0$-m' \\geq -b\u00a0\\geq -M'$<br \/>\ndonc\u00a0$m \\leq a \\leq M$<br \/>\n$-M' \\leq - b \\leq -m'$<br \/>\nPar cons\u00e9quent, $m -M'\u00a0\\leq a -\u00a0b \\leq M -m'$<\/li>\n<li>$R_5$ :\u00a0$a,b, m,m',M,M'$ \u00e9tant\u00a0des \u00a0nombres r\u00e9els<br \/>\nSi $0 \\leq m \\leq a \\leq M$<br \/>\nSi $0 \\leq m' \\leq\u00a0b\u00a0\\leq M'$<br \/>\nalors\u00a0$m + m' \\leq a b \\leq M + M'$<\/li>\n<li>$R_6$ :<\/li>\n<li>$R_7$ :<\/li>\n<li>$R_8$ :<\/li>\n<li>$R_9$ :<\/li>\n<li>$R_{10}$ :<\/li>\n<li>$R_{11}$ :<\/li>\n<li>$R_{12}$ :<\/li>\n<li>$R_{13}$ :<\/li>\n<li><\/li>\n<\/ul>\n<\/li>\n<li>Aider l'\u00e9l\u00e8ve \u00e0 ma\u00eetriser les expressions alg\u00e9briques :<\/li>\n<li><span style=\"color: #ff0000;\"><strong>Ma\u00eetriser la notion de valeur absolue d'un nombre r\u00e9el :<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-520 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/femme1.gif\" alt=\"\" width=\"133\" height=\"120\" \/><span style=\"color: #0000ff;\"><b>Retenons :<br \/>\n<\/b><\/span><\/strong><\/span><strong><span style=\"color: #008000;\">La valeur absolue d'un nombre est sa distance \u00e0 $0$ :\u00a0$\\mid x \\mid = d(x ; 0)$<\/span><\/strong><strong><span style=\"color: #008000;\">Pour d\u00e9terminer\u00a0la valeur absolue d'un nombre x, on utilise un algorithme :<\/span><\/strong><strong><span style=\"color: #008000;\">On d\u00e9termine d'abord le \u00a0signe de ce nombre<\/span><\/strong><span style=\"color: #008000;\"><b>Si ce signe est positif alors la valeur absolue de ce nombre\u00a0est ce nombre.<\/b><\/span><span style=\"color: #008000;\"><b>Si ce signe est n\u00e9gatif alors la valeur absolue de ce nombre\u00a0 est l'oppos\u00e9 de ce nombre.<br \/>\n<\/b><\/span><span style=\"color: #0000ff;\"><b>Exercice 1<br \/>\n<\/b><\/span>D\u00e9terminer les valeurs absolues des nombres suivants :56 ; - 75 ; 3.14 \u00a0 ; $x^2$\u00a0;\u00a0$-x^4$<br \/>\n<span style=\"color: #0000ff;\"><b>Retenons :<br \/>\n<\/b><\/span><b><span style=\"color: #008000;\">Voici la courbe d'\u00e9quation y =| x |\u00a0<\/span><\/b><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1408 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs-300x199.png\" alt=\"\" width=\"300\" height=\"199\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs-300x199.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs-768x508.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs-1024x678.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs.png 1112w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><span style=\"color: #0000ff;\"><b>Application aux \u00e9quations<br \/>\n<\/b><\/span><span style=\"color: #0000ff;\"><b>Retenons\u00a0:<\/b><\/span><span style=\"color: #008000;\"><b> <img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1403 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs1-300x136.png\" alt=\"\" width=\"439\" height=\"199\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs1-300x136.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs1-768x348.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs1.png 948w\" sizes=\"auto, (max-width: 439px) 85vw, 439px\" \/><\/b><\/span><span style=\"color: #008000;\"><b>1\u00b0) Si $a &lt; 0$ alors l'\u00e9quation $|x| = a$ n'a aucune\u00a0solution car la valeur absolue d'un nombre est toujours\u00a0positive ou nulle<\/b><\/span><span style=\"color: #008000;\"><b>2\u00b0) Si $a &gt; 0$ alors l'\u00e9quation $|x| = a$ admet pour solutions $-a$ et $a$\u00a0car deux nombres oppos\u00e9s\u00a0ont m\u00eame valeur absolue\u00a0<\/b><\/span><span style=\"color: #008000;\"><b>3\u00b0) Si $a = 0$ alors l'\u00e9quation $|x| = a$ admet pour solution unique $0$ car la valeur absolue d'un nombre est nulle si et seulement si ce nombre est nul.<\/b><\/span><span style=\"color: #008000;\"><b><span style=\"color: #0000ff;\">Exercice<\/span>\u00a0<\/b><\/span>R\u00e9soudre les \u00e9quations suivantes d'inconnue x :a) $|x| = -3$<\/p>\n<p>b) $|x| = 7$<\/p>\n<p>c) $|x - 3| = -3$<\/p>\n<p>d) $|x - 3| = 0$<\/p>\n<p><b><span style=\"color: #0000ff;\">Retenons :<\/span><\/b><\/p>\n<p><b> <span style=\"color: #008000;\">La distance entre des nombres $a$ et $b$ est la valeur absolue de leur diff\u00e9rence.<\/span><\/b><\/p>\n<p><span style=\"color: #008000;\">On \u00e9crira $d(a;b) = | a - b |$<\/span><\/p>\n<p><strong><span style=\"color: #0000ff;\">Exercice<br \/>\n<\/span><\/strong><br \/>\n1\u00b0)Calculer $d(3;4) ; d( 4;-7) ; d(-7;-11)$<\/p>\n<p>2\u00b0) R\u00e9soudre en utilisant la notion de distance l'\u00e9quation : $|x - 3| = 2$<\/p>\n<p><span style=\"color: #0000ff;\"><b>Retenons :<\/b><\/span><\/p>\n<p><span style=\"color: #008000;\"><b>1\u00b0)Si $a &lt;= 0$ alors $<\/b><b>|nombre|&lt; a$<\/b><b> est impossible\u00a0<\/b><\/span><\/p>\n<p><span style=\"color: #008000;\"><b>2\u00b0)Si $a &gt; 0$ alors $<\/b><b>|nombre| &lt; a$<\/b><b> signifie que ce nombre est strictement compris entre $- a$ et $a$<\/b><\/span><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1404 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs2-300x168.png\" alt=\"\" width=\"231\" height=\"129\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs2-300x168.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs2-768x429.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs2.png 924w\" sizes=\"auto, (max-width: 231px) 85vw, 231px\" \/><\/p>\n<p><span style=\"color: #008000;\"><b>3\u00b0)Si $a &lt; 0$ alors $<\/b><b>|nombre|&gt; a$\u00a0<\/b><b>est vraie pour tout nombre.<\/b><\/span><\/p>\n<p><span style=\"color: #008000;\"><b>4\u00b0)Si $a &gt;= 0$ alors $<\/b><b>|nombre| &gt; a$\u00a0<\/b><b>signifie que ce nombre est strictement plus petit que $-a$ ou strictement plus grand que $a$.<\/b><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1405 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs3-300x177.png\" alt=\"\" width=\"300\" height=\"177\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs3-300x177.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs3-768x454.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs3.png 816w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1406 aligncenter\" src=\"http:\/\/www2.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs4-300x181.png\" alt=\"\" width=\"300\" height=\"181\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs4-300x181.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs4-768x464.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/abs4.png 804w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<li><strong><span style=\"color: #ff0000;\">Ma\u00eetriser la notion de partie enti\u00e8re d'un nombre r\u00e9el :<\/span><\/strong><strong><br \/>\n<span style=\"color: #0000ff;\">Retenons :\u00a0<\/span><br \/>\n<span style=\"color: #339966;\">Tout nombre\u00a0r\u00e9el $x$ est compris entre 2 entiers relatifs cons\u00e9cutifs $n$ et $n + 1$.<br \/>\n<\/span><span style=\"color: #339966;\">On a donc\u00a0$\\forall x \\ in R \\qquad n \\leq x &lt; n + 1$<\/span><\/strong><br \/>\n<strong><strong><span style=\"color: #339966;\">Cet entier relatif $n$ s'appelle la partie enti\u00e8re de $x$ et se note $[x]$.<\/span><br \/>\n<span style=\"color: #339966;\"> La partie enti\u00e8re d'un r\u00e9el est donc le plus grand de tous les entiers relatifs qui le pr\u00e9c\u00e8dent.<\/span><br \/>\n<span style=\"color: #339966;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1662 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/entx-300x229.png\" alt=\"\" width=\"300\" height=\"229\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/entx-300x229.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/entx-768x587.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/entx.png 824w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/span><span style=\"color: #339966;\">Comme $[x] \\leq x &lt; [x] +1$ alors $0 \\leq x - [x] &lt; 1$.<\/span><br \/>\n<span style=\"color: #339966;\">$x \\mapsto x - [x]$ s'appelle la fonction partie\u00a0fractionnaire de $x$.\u00a0C'est une fonction p\u00e9riodique de p\u00e9riode $1$<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1663 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/epix-291x300.png\" alt=\"\" width=\"291\" height=\"300\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/epix-291x300.png 291w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/epix.png 574w\" sizes=\"auto, (max-width: 291px) 85vw, 291px\" \/><\/span><br \/>\n<\/strong><\/strong><span style=\"color: #0000ff;\"><strong>Entrainement :<\/strong><\/span><br \/>\nSujet sur la partie enti\u00e8re :\u00a0<a href=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/partieentiere.pdf\">partieentiere<\/a><br \/>\nCorrig\u00e9 : \u00a0<a href=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/04\/partieentierecor.pdf\">partieentierecor<\/a><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>R\u00e9sultats de base \u00e0 conna\u00eetre par coeur\u00a0 8 identit\u00e9s remarquables dans l'ensemble des r\u00e9els(mieux dans tout anneau commutatif) : $IR_1$ : $(a +b)^2 = a^2 + 2ab + b^2$ d\u00e9monstration : $(a +b)^2 =(a + b)(a + b)\u00a0= a^2 \u00a0+ab + ab + b^2 =a^2 + 2ab + b^2$ Le carr\u00e9 d'une somme est la &hellip; <a href=\"https:\/\/www.mathnique.com\/site\/methodes\/\" class=\"more-link\">Continuer la lecture<span class=\"screen-reader-text\"> de &laquo;&nbsp;M\u00e9thodes et Outils&nbsp;&raquo;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-1158","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/comments?post=1158"}],"version-history":[{"count":44,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1158\/revisions"}],"predecessor-version":[{"id":2403,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/1158\/revisions\/2403"}],"wp:attachment":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/media?parent=1158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}