Wzory skr\u00f3conego mno\u017cenia<\/p>\n\n\n
Wzory skr\u00f3conego mno\u017cenia to zestaw wyra\u017ce\u0144 algebraicznych, kt\u00f3re upraszczaj\u0105 przekszta\u0142canie r\u00f3wna\u0144 i wyra\u017ce\u0144 matematycznych.<\/p>\n\r\n<\/div>\n\n\n
Kwadrat sumy<\/p>\n\n\n
Kwadrat sumy dw\u00f3ch wyra\u017ce\u0144 mo\u017cna przedstawi\u0107 jako sum\u0119 kwadrat\u00f3w tych wyra\u017ce\u0144 oraz ich podwojonego iloczynu: \\[(a + b)^2 = a^2 + 2ab + b^2\\]<\/p>\n\r\n<\/div>\n\n
\\[(x + 3)^2 = x^2 + 2 \\cdot x \\cdot 3 + 3^2 = x^2 + 6x + 9\\]<\/p>\n\r\n<\/div>\n\n\n
Kwadrat r\u00f3\u017cnicy<\/p>\n\n\n
Kwadrat r\u00f3\u017cnicy dw\u00f3ch wyra\u017ce\u0144 mo\u017cna przedstawi\u0107 jako sum\u0119 kwadrat\u00f3w tych wyra\u017ce\u0144 oraz r\u00f3\u017cnicy ich podwojonego iloczynu: \\[(a – b)^2 = a^2 – 2ab + b^2\\]<\/p>\n\r\n<\/div>\n\n
\\[(x – 4)^2 = x^2 – 2 \\cdot x \\cdot 4 + 4^2 = x^2 – 8x + 16\\]<\/p>\n\r\n<\/div>\n\n\n
R\u00f3\u017cnica kwadrat\u00f3w<\/p>\n\n\n
R\u00f3\u017cnica kwadrat\u00f3w dw\u00f3ch wyra\u017ce\u0144 jest r\u00f3wna iloczynowi sumy i r\u00f3\u017cnicy tych wyra\u017ce\u0144: \\[a^2 – b^2 = (a – b)(a + b)\\]<\/p>\n\r\n<\/div>\n\n
\\[x^2 – 9 = (x – 3)(x + 3)\\]<\/p>\n\r\n<\/div>\n\n\n
Sze\u015bcian sumy<\/p>\n\n\n
Sze\u015bcian sumy dw\u00f3ch wyra\u017ce\u0144 mo\u017cna przedstawi\u0107 jako sum\u0119 sze\u015bcian\u00f3w tych wyra\u017ce\u0144 oraz trzykrotno\u015bci iloczynu ich sumy i kwadrat\u00f3w: \\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\\]<\/p>\n\r\n<\/div>\n\n
\\[(x + 2)^3 = x^3 + 6x^2 + 12x + 8\\]<\/p>\n\r\n<\/div>\n\n\n
Sze\u015bcian r\u00f3\u017cnicy<\/p>\n\n\n
Sze\u015bcian r\u00f3\u017cnicy dw\u00f3ch wyra\u017ce\u0144 mo\u017cna przedstawi\u0107 jako r\u00f3\u017cnic\u0119 sze\u015bcian\u00f3w tych wyra\u017ce\u0144 oraz trzykrotno\u015bci iloczynu ich r\u00f3\u017cnicy i kwadrat\u00f3w: \\[(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3\\]<\/p>\n\r\n<\/div>\n\n
\\[(x – 1)^3 = x^3 – 3x^2 + 3x – 1\\]<\/p>\n\r\n<\/div>\n\n\n
Suma sze\u015bcian\u00f3w<\/p>\n\n\n
Suma sze\u015bcian\u00f3w dw\u00f3ch wyra\u017ce\u0144 jest r\u00f3wna iloczynowi sumy tych wyra\u017ce\u0144 oraz kwadratu r\u00f3\u017cnicy: \\[a^3 + b^3 = (a + b)(a^2 – ab + b^2)\\]<\/p>\n\r\n<\/div>\n\n
\\[x^3 + 8 = (x + 2)(x^2 – 2x + 4)\\]<\/p>\n\r\n<\/div>\n\n\n
R\u00f3\u017cnica sze\u015bcian\u00f3w<\/p>\n\n\n
R\u00f3\u017cnica sze\u015bcian\u00f3w dw\u00f3ch wyra\u017ce\u0144 jest r\u00f3wna iloczynowi r\u00f3\u017cnicy tych wyra\u017ce\u0144 oraz kwadratu ich sumy: \\[a^3 – b^3 = (a – b)(a^2 + ab + b^2)\\]<\/p>\n\r\n<\/div>\n\n
\\[x^3 – 27 = (x – 3)(x^2 + 3x + 9)\\]<\/p>\n\r\n<\/div>\n\n\n
R\u00f3\u017cnica n-tych pot\u0119g<\/p>\n\n\n
R\u00f3\u017cnica n-tych pot\u0119g dw\u00f3ch wyra\u017ce\u0144 mo\u017cna wyrazi\u0107 jako iloczyn r\u00f3\u017cnicy tych wyra\u017ce\u0144 oraz szeregu pot\u0119g ich sum i r\u00f3\u017cnic: \\[a^n – b^n = (a – b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + … + ab^{n-2} + b^{n-1})\\]W szczeg\u00f3lno\u015bci: \\[a^n – 1 = (a – 1)(a^{n-1} + a^{n-2} + … + a + 1)\\]<\/p>\n\r\n<\/div>\n\n
\\[x^4 – 16 = (x^2 – 4)(x^2 + 4)\\]<\/p>\n\r\n<\/div>","protected":false},"excerpt":{"rendered":"
Wzory skr\u00f3conego mno\u017cenia Kwadrat sumy Kwadrat r\u00f3\u017cnicy R\u00f3\u017cnica kwadrat\u00f3w Sze\u015bcian sumy Sze\u015bcian r\u00f3\u017cnicy Suma sze\u015bcian\u00f3w R\u00f3\u017cnica sze\u015bcian\u00f3w R\u00f3\u017cnica n-tych pot\u0119g<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[106],"tags":[107,163],"class_list":["post-151","post","type-post","status-publish","format-standard","hentry","category-matematyka","tag-matematyka","tag-wzory-skroconego-mnozenia"],"_links":{"self":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/151","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/comments?post=151"}],"version-history":[{"count":3,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/151\/revisions"}],"predecessor-version":[{"id":758,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/151\/revisions\/758"}],"wp:attachment":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/media?parent=151"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/categories?post=151"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/tags?post=151"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}