\n 1. Synthetic Division Steps<\/a>\n
\n\t 2. Synthetic Division Examples<\/a>\n\t
\n\t 3. Polynomial Remainder Theorem<\/a><\/font>\n <\/div>\n
\n
Moving on from the box method<\/i><\/a> for division of polynomials, another approach that can be used at times is ‘synthetic division’.\n On this Page: 1. Synthetic Division Steps 2. Synthetic Division Examples 3. Polynomial Remainder Theorem Moving on from the box method for division of polynomials, another approach that can be used at times is ‘synthetic division’. This approach can be used when a polynomial is to be divided by a linear expression with a leading… Read More »Synthetic Division of Polynomials Examples<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"neve_meta_sidebar":"","neve_meta_container":"","neve_meta_enable_content_width":"","neve_meta_content_width":0,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"wf_page_folders":[14],"class_list":["post-807","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/comments?post=807"}],"version-history":[{"count":17,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/807\/revisions"}],"predecessor-version":[{"id":3609,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/807\/revisions\/3609"}],"wp:attachment":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/media?parent=807"}],"wp:term":[{"taxonomy":"wf_page_folders","embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/wf_page_folders?post=807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nThis approach can be used when a polynomial is to be divided by a linear expression with a leading coefficient of 1. The synthetic division of polynomials examples on this page will show the steps to take in such situations.<\/p>\n
\n<\/a>\n
\n\n\n\nSynthetic Division Steps:<\/span><\/h2>\n\n\n\n
\nWe will introduce the first of our synthetic division of polynomials examples by looking at the division of the polynomial 2x^3 + 4x^2 \\space {\\text{--}} \\space 3x \\space {\\text{--}} \\space 3<\/span>,\n
\nby the linear expression x \\space {\\text{--}} \\space 1<\/span>.\n
\n1)<\/b><\/u><\/font><\/font>\n
\nSketch half a box, with a left side and a base.\n
\nThen write the coefficients and constant of the polynomial along the top of the half box.<\/font> \n
\n\n\n\n![\"First](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/first-box-synth-div.png\")
<\/figure>\n\n\n\n
\n2)<\/b><\/u><\/font><\/font>\n
\nSolve for x<\/span> in the linear expression.\n
\nWrite this number on the left side of the half box, while also bringing the first coefficient down underneath.<\/font>\n
\n\n\n\n![\"Bring](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/second-box-synth-div.png\")
<\/figure>\n\n\n\n
\n3)<\/b><\/u><\/font><\/font>\n
\nNow multiply this first coefficient underneath by the value we found and placed on the left.\n
\nThen place this multiplication result at he base of the next column.<\/font>\n
\n\n\n\n![\"Multiply](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/third-box-synth-div.png\")
<\/figure>\n\n\n\n
\n4)<\/b><\/u><\/font><\/font>\n
\nNow add the numbers in this next column together, and place the result underneath the column.<\/font>\n
\n\n\n\n![\"Add](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/fourth-box-synth-div.png\")
<\/figure>\n\n\n\n
\n5)<\/b><\/u><\/font><\/font>\n
\nThe repeat the same steps for the remaining columns of the half box, until there are no more coefficient columns left to do.<\/font>\n
\n\n\n\n![\"Complete](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/synthetic-division-of-polynomials-examples.png\")
<\/figure>\n\n\n\n
\nWhen we are done, the numbers underneath to half box will be the answer to the polynomial division.\n
\nThe degree of the answer is one less than the dividend, so our answer is 2x^2 + 6x + 3<\/span>.\n
\nWhen 0 is the final number of the computations involved, there is no remainder to the division sum.<\/font>\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nSynthetic Division of Polynomials
Examples<\/span><\/h2>\n\n\n\n
\n(1.1) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\n( 2x^3 + 6x^2 + 2x \\space {\\text{--}} \\space 5 ) \\space \\div \\space ( x + 1 )<\/span><\/font>\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\nx + 1 = 0 \\space \\space => \\space \\space x = {\\text{-}}1<\/span><\/font>\n
\n\n\n\n![\"Division](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/synth-div-example-1.png\")
<\/figure>\n\n\n\n
\n( 2x^3 + 6x^2 + 2x \\space {\\text{--}} \\space 5 ) \\space \\div \\space ( x + 1 ) \\space = \\space 2x^2 + 4x \\space {\\text{-}} \\space 2 + \\frac{3}{x + 1}<\/span>\n
\n(1.2) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\n( 3x^3 + 6x^2 \\space {\\text{--}} \\space 3x + 2 ) \\space \\div \\space ( x + 2 )<\/span><\/font>\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\nx + 2 = 0 \\space \\space => \\space \\space x = {\\text{-}}2<\/span><\/font>\n
\n\n\n\n![\"Sums](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/synth-div-ex-2.png\")
<\/figure>\n\n\n\n
\n( 3x^3 + 6x^2 \\space {\\text{--}} \\space 3x + 2 ) \\space \\div \\space ( x + 2 ) \\space = \\space 3x^2 \\space {\\text{--}} \\space 3 + \\frac{8}{x + 2}<\/span>\n
\n(1.3) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\n( 2x^3 + 5x^2 \\space {\\text{--}} \\space 9x + 4 ) \\space \\div \\space ( 2x \\space {\\text{--}} \\space 1 )<\/span><\/font>\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\nHere the leading coefficient of the divisor is not 1, it is 2.\n
\nSo we have to make the 2 into a 1 by dividing through by 2.<\/font>\n
\n( 2x \\space {\\text{--}} \\space 1 ) \\div 2 \\space = \\space x \\space {\\text{--}} \\space \\frac{1}{2}<\/span>\n
\nx \\space {\\text{--}} \\space \\frac{1}{2} = 0 \\space \\space => \\space \\space x = \\frac{1}{2}<\/span><\/font>\n
\n\n\n\n![\"Example](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/sd-ex-3.png\")
<\/figure>\n\n\n\n
\n( 2x^3 + 5x^2 \\space {\\text{--}} \\space 9x + 4 ) \\space \\div \\space ( 2x \\space {\\text{--}} \\space 1 ) \\space = \\space 2x^2 + 6x \\space {\\text{--}} \\space 6 + \\frac{1}{2x \\space {\\text{--}} \\space 1}<\/span>\n
\n(1.4) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\n( x^3 + 4x + 3 ) \\space \\div \\space ( x + 2 )<\/span><\/font>\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\nx + 2 = 0 \\space \\space => \\space \\space x = {\\text{-}}2<\/span><\/font>\n
\nIn this example there is no x^2<\/span> term in the larger polynomial being divided.\n
\nWe don’t skip the entry out, instead we place a 0 above, and include it in the synthetic division steps.<\/font>\n
\n\n\n\n![\"Example](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/sd-ex-3.5.png\")
<\/figure>\n\n\n\n
\n( x^3 + 4x + 3 ) \\space \\div \\space ( x + 2 ) \\space = \\space x^2 \\space {\\text{--}} \\space 2x + 8 \\space {\\text{--}} \\space \\frac{19}{x + 2}<\/span>\n
\n(1.5) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\nIf ( x^3 + ax^2 \\space {\\text{--}} \\space 4x + 4 ) \\space \\div \\space ( x \\space {\\text{--}} \\space 1 )<\/span><\/font> has a remainder of 3.\n
\nWhat is the value of a<\/span>?\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\nx \\space {\\text{--}} \\space 1 = 0 \\space \\space => \\space \\space x = 1<\/span><\/font>\n
\nWe can proceed by placing a<\/i><\/font> in the correct position above, and following the usual steps.<\/font>\n
\n\n\n\n![\"Finding](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/sd-ex-4.png\")
<\/figure>\n\n\n\n
\n1 + a = 3<\/span> => 1 + a \\space {\\text{--}} \\space 1 = 3 \\space {\\text{--}} \\space 1<\/span> => a = 2<\/span>\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nPolynomial Remainder Theorem<\/span><\/h2>\n\n\n\n
\nRelated to synthetic division of polynomials examples is something called the ‘remainder theorem’.\n
\nWhich is that when a polynomial f(x)<\/span> is divided by ( x \\space {\\text{--}} \\space h )<\/span>,\n
\nthe remainder will be f(h)<\/span>. \n
\nIf we remember example (1.2), the remainder was 8, when dividing by ( x + 2 )<\/span>.\n
\nSo by the remainder theorem, we expect f({\\text{-}}2)<\/span> to equal 8.\n
\nf(x) \\space = \\space 3x^3 + 6x^2 \\space {\\text{--}} \\space 3x + 2<\/span>\n
\nf({\\text{-}}2) \\space = \\space 3({\\text{-}}2)^3 + 6({\\text{-}}2)^2 \\space {\\text{--}} \\space 3({\\text{-}}2) + 2<\/span>\n
\n= \\space {\\text{-}}24 + 24 + 6 + 2 \\space = \\space 8<\/span>\n
\nThis can be a useful way of checking if you have done the correct steps when doing a synthetic division sum.\n
\n\n\n\n
\n\n\n\n
\n
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