{"id":10853,"date":"2017-06-05T21:31:44","date_gmt":"2017-06-05T21:31:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10853"},"modified":"2020-10-22T09:30:03","modified_gmt":"2020-10-22T09:30:03","slug":"summary-dividing-monomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/summary-dividing-monomials\/","title":{"raw":"12.3.d - Summary: Dividing Polynomials","rendered":"12.3.d &#8211; Summary: Dividing Polynomials"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>\r\n<p id=\"fs-id1165135393414\"><strong>Given two polynomials, where the divisor is in the form [latex]x-k[\/latex], how to use synthetic division to divide<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135393418\">\r\n \t<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Repeat steps\u00a0[latex]5[\/latex] and\u00a0[latex]6[\/latex] for the remaining columns.<\/li>\r\n \t<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree\u00a0[latex]0[\/latex]. The next number from the right has degree\u00a0[latex]1[\/latex], and the next number from the right has degree\u00a0[latex]2[\/latex], and so on.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<ul id=\"eip-584\">\r\n \t<li><strong>Equivalent Fractions Property<\/strong>\r\n<ul id=\"eip-id1170323872699\">\r\n \t<li id=\"eip-id1170322925692\">If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}[\/latex] and [latex]{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Zero Exponent<\/strong>\r\n<ul id=\"eip-id1170320472334\">\r\n \t<li>If [latex]a[\/latex] is a non-zero number, then [latex]{a}^{0}=1[\/latex].<\/li>\r\n \t<li>Any nonzero number raised to the zero power is [latex]1[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Quotient Property for Exponents<\/strong>\r\n<ul id=\"eip-id1170324175453\">\r\n \t<li id=\"eip-id1170321647368\">If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then\r\n[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m&gt;n[\/latex] and [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n&gt;m[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Quotient to a Power Property for Exponents<\/strong>\r\n<ul id=\"eip-id1170326477710\">\r\n \t<li>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then\r\n[latex]{\\Large{\\left(\\frac{a}{b}\\right)}}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}}[\/latex]<\/li>\r\n \t<li>To raise a fraction to a power, raise the numerator and denominator to that power.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<p class=\"hanging-indent\"><strong>Synthetic Division\u00a0 \u00a0<\/strong>Synthetic division is a shortcut that can be used when the divisor is a binomial in the form\u00a0\u00a0[latex]x\u2013k[\/latex]<em style=\"font-size: 1rem;text-align: initial\">, <\/em><span style=\"font-size: 1rem;text-align: initial\">for a real number\u00a0[latex]k[\/latex].\u00a0In <\/span><strong style=\"font-size: 1rem;text-align: initial\">synthetic division<\/strong><span style=\"font-size: 1rem;text-align: initial\">, only the coefficients are used in the division process.<\/span><\/p>\r\n\r\n<h2><\/h2>\r\n<h2><\/h2>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>\n<p id=\"fs-id1165135393414\"><strong>Given two polynomials, where the divisor is in the form [latex]x-k[\/latex], how to use synthetic division to divide<\/strong><\/p>\n<ol id=\"fs-id1165135393418\">\n<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Repeat steps\u00a0[latex]5[\/latex] and\u00a0[latex]6[\/latex] for the remaining columns.<\/li>\n<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree\u00a0[latex]0[\/latex]. The next number from the right has degree\u00a0[latex]1[\/latex], and the next number from the right has degree\u00a0[latex]2[\/latex], and so on.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<ul id=\"eip-584\">\n<li><strong>Equivalent Fractions Property<\/strong>\n<ul id=\"eip-id1170323872699\">\n<li id=\"eip-id1170322925692\">If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<br \/>\n[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}[\/latex] and [latex]{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Zero Exponent<\/strong>\n<ul id=\"eip-id1170320472334\">\n<li>If [latex]a[\/latex] is a non-zero number, then [latex]{a}^{0}=1[\/latex].<\/li>\n<li>Any nonzero number raised to the zero power is [latex]1[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Quotient Property for Exponents<\/strong>\n<ul id=\"eip-id1170324175453\">\n<li id=\"eip-id1170321647368\">If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then<br \/>\n[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n[\/latex] and [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n>m[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Quotient to a Power Property for Exponents<\/strong>\n<ul id=\"eip-id1170326477710\">\n<li>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then<br \/>\n[latex]{\\Large{\\left(\\frac{a}{b}\\right)}}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}}[\/latex]<\/li>\n<li>To raise a fraction to a power, raise the numerator and denominator to that power.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p class=\"hanging-indent\"><strong>Synthetic Division\u00a0 \u00a0<\/strong>Synthetic division is a shortcut that can be used when the divisor is a binomial in the form\u00a0\u00a0[latex]x\u2013k[\/latex]<em style=\"font-size: 1rem;text-align: initial\">, <\/em><span style=\"font-size: 1rem;text-align: initial\">for a real number\u00a0[latex]k[\/latex].\u00a0In <\/span><strong style=\"font-size: 1rem;text-align: initial\">synthetic division<\/strong><span style=\"font-size: 1rem;text-align: initial\">, only the coefficients are used in the division process.<\/span><\/p>\n<h2><\/h2>\n<h2><\/h2>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10853\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21046,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"85a435f83bbe4b7c84f32b954269e6c1","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10853","chapter","type-chapter","status-publish","hentry"],"part":8336,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10853","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/21046"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10853\/revisions"}],"predecessor-version":[{"id":20403,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10853\/revisions\/20403"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8336"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10853\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10853"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10853"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10853"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=10853"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}