{"id":86,"date":"2017-08-23T07:43:53","date_gmt":"2017-08-23T07:43:53","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=86"},"modified":"2017-09-18T15:34:30","modified_gmt":"2017-09-18T15:34:30","slug":"factoring-polynomials","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/factoring-polynomials\/","title":{"rendered":"Factoring Polynomials"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/simplifying-rational-polynomials\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Factoring Polynomials<\/h1>\n<h4>Objective<\/h4>\n<p>Starting with a polynomial in standard form, you will study how to change a polynomial equation into a product of its\u00a0individual terms.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The different characteristics\u00a0and classifications of <strong><em>polynomials<\/em><\/strong>, and\u00a0how to determine the degree and number of terms in our\u00a0classification.<\/li>\n<li>The operations of addition, subtraction, multiplication,\u00a0and division with the goal of combining multiple polynomials into\u00a0one final equation in <strong><em>standard form<\/em><\/strong>.<\/li>\n<\/ul>\n<h3>How are polynomials factored?<\/h3>\n<p>We will study the four methods of factoring, in addition to the\u00a0preferred &#8220;trial and error&#8221; method:<\/p>\n<ol start=\"1\">\n<li>factoring by using the distributive property;<\/li>\n<li>factoring quadratic trinomials in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image002.gif\" width=\"64\" height=\"14\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/>\u00a0;<\/li>\n<li>factoring quadratic trinomials in the form <strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image004.gif\" width=\"72\" height=\"14\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/strong>;<strong>\u00a0<\/strong>and<\/li>\n<li>factoring in special cases.<\/li>\n<\/ol>\n<h4><strong>First method\u2014the distributive property <\/strong><\/h4>\n<p>Polynomials can be factored by applying the distributive\u00a0property by pulling out the common term of the polynomial. First\u00a0find the <abbr title=\"the largest factor that can be extracted from each term. The GCF can be a constant,variable, or a combination.\">greatest\u00a0common factor (GCF)<\/abbr>.<\/p>\n<p>For example, first find the GCF of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image006.gif\" width=\"59\" height=\"14\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/>\u00a0.<\/p>\n<p>Each term can be written as a product of individual terms:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image008.gif\" width=\"103\" height=\"62\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Remove the GCF from each term. Factoring out the GCF from each\u00a0term the polynomial is factored as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image010.gif\" width=\"78\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>We can verify our answer by distributing the 7<em>x<\/em> across both terms in parentheses to reproduce our original equation <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image012.gif\" width=\"69\" height=\"11\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/>\u00a0.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct factorization of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image014.gif\" width=\"104\" height=\"14\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>\u00a0by factoring out the GCF?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image016.gif\" width=\"96\" height=\"14\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image018.gif\" width=\"114\" height=\"14\" name=\"graphics11\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image020.gif\" width=\"98\" height=\"14\" name=\"graphics12\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image022.gif\" width=\"106\" height=\"14\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C. To find the GCF, expand each term.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image024.gif\" width=\"115\" height=\"64\" name=\"graphics14\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">From this, we can see that\u00a0the GCF is 2<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0which can be factored out of each term in the original\u00a0polynom<\/span>ial.<\/p>\n<\/div>\n<\/section>\n<h4>Second Method \u2014 Factoring quadratic trinomials in\u00a0the form x<sup>2<\/sup> + bx + c<\/h4>\n<p>Multiplying two binomials using the FOIL method results in a\u00a0polynomial in standard form.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image027.gif\" width=\"309\" height=\"17\" name=\"graphics16\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>To make the process easier, group the coefficients together as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image029.gif\" width=\"35\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0for the linear term <span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">rt\u00a0<\/span><\/em><span style=\"text-decoration: none;\">for th<\/span>e constant term.<\/p>\n<p><span style=\"text-decoration: none;\">Comparing this to the\u00a0standard form, we can see that the coefficient <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">on the linear term is equal to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image031.gif\" width=\"36\" height=\"14\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The constant <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is equal to <\/span><em><span style=\"text-decoration: none;\">rt<\/span><\/em><span style=\"text-decoration: none;\">. We now have two equations with two unknowns. Now find two\u00a0numbers that, when added, result in <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and when\u00a0multiplied, result in <\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\">Trial and error becomes important\u00a0in this step. Try different numbers, both positive and negative,\u00a0that will satisfy these requirements.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which are the factors of the trinomial\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image033.gif\" width=\"65\" height=\"14\" name=\"graphics19\" align=\"absmiddle\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image035.gif\" width=\"81\" height=\"14\" name=\"graphics20\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image037.gif\" width=\"80\" height=\"14\" name=\"graphics21\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image039.gif\" width=\"80\" height=\"14\" name=\"graphics22\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image041.gif\" width=\"80\" height=\"14\" name=\"graphics23\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct answer is D.\u00a0In this equation, <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">t\u00a0<\/span><\/em><span style=\"text-decoration: none;\">must add to equal 5, and they must multiply to equal 6. <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image043.gif\" width=\"50\" height=\"35\" name=\"graphics24\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">By trial and error (and\u00a0inspection), or solving for each variable, we can conclude that <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">t\u00a0<\/span><\/em><span style=\"text-decoration: none;\">equal 3 and 2. Then, simply write the factors\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p2_clip_image045.gif\" width=\"80\" height=\"14\" name=\"graphics25\" align=\"absmiddle\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<\/div>\n<\/section>\n<h3>How do we know a trinomial is factored correctly?<\/h3>\n<p>If we use FOIL and multiply the two binomials (assuming the\u00a0factors are correct), we will reproduce the original trinomial\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image002.gif\" width=\"65\" height=\"14\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/>.\u00a0<span style=\"text-decoration: none;\">If not, there was an error in\u00a0the selection of <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and the process must be repeated until the correct numbers ar<\/span>e\u00a0found.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>It is very important to take a few extra minutes to check that you\u00a0have selected the correct factors. Factoring more complex trinomials can be difficult, and using FOIL to\u00a0check your answer is an important step that must not be overlooked.<\/p>\n<\/div>\n<h4>Third Method \u2014 Factoring quadratic trinomials in\u00a0the form <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image004.gif\" width=\"72\" height=\"14\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/><\/h4>\n<p><span style=\"text-decoration: none;\">Now there is a coefficient\u00a0on the quadratic term equal to <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and a third equation must be added to help find its value. With\u00a0the addition of this coefficient, factoring the quadratic equation\u00a0will produce two binomials in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image006.gif\" width=\"96\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p><span style=\"text-decoration: none;\">The equations become <\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image008.gif\" width=\"43\" height=\"7\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image010.gif\" width=\"80\" height=\"14\" name=\"graphics7\" align=\"ABSBOTTOM\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image012.gif\" width=\"34\" height=\"9\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/>.\u00a0From here, the goal is to find <\/span><em><span style=\"text-decoration: none;\">m<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">r,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0Remember that, as long as the trinomial is not too difficult, you\u00a0can probably use trial and error to find the solution. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which is the correct factorization of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image014.gif\" width=\"79\" height=\"14\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image016.gif\" width=\"85\" height=\"14\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image018.gif\" width=\"85\" height=\"14\" name=\"graphics11\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image020.gif\" width=\"85\" height=\"14\" name=\"graphics12\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image022.gif\" width=\"77\" height=\"14\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. The following information is known:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image024.gif\" width=\"84\" height=\"59\" name=\"graphics14\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">We see that <\/span><em><span style=\"text-decoration: none;\">m\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 2, <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 1, and <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 1 and t = 5. In factored form,<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image026.gif\" width=\"182\" height=\"17\" name=\"graphics15\" align=\"absmiddle\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h4>Fourth method \u2014 Factoring special cases<\/h4>\n<p>Sometimes polynomials that initially seem complex can be\u00a0factored easily by noticing special cases.<\/p>\n<p><em>Difference of two squares<\/em> :\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image028.gif\" width=\"142\" height=\"17\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p><em>Perfect Square Trinomials<\/em> :\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image030.gif\" width=\"145\" height=\"43\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which shows the correct factorization of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image032.gif\" width=\"79\" height=\"11\" name=\"graphics18\" align=\"absmiddle\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image034.gif\" width=\"45\" height=\"14\" name=\"graphics19\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image036.gif\" width=\"52\" height=\"14\" name=\"graphics20\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image038.gif\" width=\"52\" height=\"14\" name=\"graphics21\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image040.gif\" width=\"45\" height=\"14\" name=\"graphics22\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct answer is B.\u00a0Notice that the <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">-term\u00a0and the <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\">-term\u00a0are perfect squares. We can write this trinomial as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image042.gif\" width=\"135\" height=\"17\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which fits the perfect square trinomial formula. In factored\u00a0form, this is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image044.gif\" width=\"52\" height=\"14\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\">Expand the binomial to check the\u00a0answer.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image046.gif\" width=\"149\" height=\"14\" name=\"graphics25\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct factorization of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image048.gif\" width=\"48\" height=\"11\" name=\"graphics26\" align=\"absmiddle\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image050.gif\" width=\"96\" height=\"14\" name=\"graphics27\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image052.gif\" width=\"96\" height=\"14\" name=\"graphics28\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image054.gif\" width=\"95\" height=\"14\" name=\"graphics29\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image056.gif\" width=\"96\" height=\"14\" name=\"graphics30\" align=\"absmiddle\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. This is a difference of squares\u00a0polynomial and can be written in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image058.gif\" width=\"57\" height=\"14\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Apply the formula.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image060.gif\" width=\"170\" height=\"14\" name=\"graphics32\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Use the FOIL method to check accuracy.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p3_clip_image062.gif\" width=\"161\" height=\"14\" name=\"graphics33\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>Why is factoring important?<\/strong><\/h3>\n<p><span style=\"text-decoration: none;\">Factoring is one method\u00a0that lets us find the solution, or roots, of the polynomial. In\u00a0graphical terms, this is the point or points where the graph\u00a0crosses the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0and where the polynomial is equal to zero. <\/span><\/p>\n<p>For example, three roots of a polynomial function are \u20132,\u00a03, and 3. We can then write the following function in factored\u00a0form:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p4_clip_image002.gif\" width=\"167\" height=\"14\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Notice that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p4_clip_image004.gif\" width=\"30\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is equal to 0 if \u20132, 3, or 3 is substituted into this\u00a0equation. Use the FOIL method to combine the first two binomials.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p4_clip_image006.gif\" width=\"155\" height=\"17\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>Use the distributive property to combine this trinomial and the\u00a0third binomial.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p4_clip_image008.gif\" width=\"215\" height=\"66\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>If we draw the graph of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s2_p4_clip_image010.gif\" width=\"156\" height=\"17\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0it is evident what roots mean.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math%20Mod%203%5B1%5D.1%20Art%20001.JPG\" alt=\"Graph of polynomial\" width=\"433\" height=\"280\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/><\/center><\/p>\n<p align=\"LEFT\"><span style=\"text-decoration: none;\">On the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">&#8211;<\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">coordinate plane,\u00a0the graph touches and reflects off of the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0at the point <\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">=\u00a03, and crosses the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0at <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= -2. Because the graph reflects and does not cross the axis at <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 3, we have two identical roots at this point. Therefore, the roots for this\u00a0polynomial are located at \u20132, 3, and 3. <\/span><\/p>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/simplifying-rational-polynomials\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Factoring Polynomials Objective Starting with a polynomial in standard form, you will study how to change a polynomial equation into a product of its\u00a0individual terms. Previously Covered: The different characteristics\u00a0and classifications of polynomials, and\u00a0how to determine the degree and number of terms in our\u00a0classification. The operations of addition, subtraction, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-86","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/86","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=86"}],"version-history":[{"count":14,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/86\/revisions"}],"predecessor-version":[{"id":738,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/86\/revisions\/738"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}