{"id":94,"date":"2017-08-23T07:48:33","date_gmt":"2017-08-23T07:48:33","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=94"},"modified":"2017-09-21T19:46:42","modified_gmt":"2017-09-21T19:46:42","slug":"algebra-functions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions\/","title":{"rendered":"Algebra &#038; Functions"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><!--<a href=\"math_02_09.html\" class=\"button button-primary\">\u2b05 Previous Lesson<\/a>--><br \/>\n<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\/factoring-polynomials\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Algebra &amp; Functions<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study more advanced algebra topics such as rational polynomials, factoring, quadratics,\u00a0conics, exponents, and the Fundamental Theorem of Algebra.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><em><strong>Functions<\/strong><\/em> are based on rules that\u00a0define inputs and outputs. For each\u00a0input, there is exactly one output.<\/li>\n<li><strong><em>Composite functions<\/em><\/strong> are evaluated in\u00a0series, meaning that the output of the first\u00a0function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image002.gif\" width=\"28\" height=\"14\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/> is the input for the second function, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image004.gif\" width=\"27\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Composite functions are written as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image006.gif\" width=\"54\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li>Functions can have <strong><em>inverses<\/em><\/strong>.\u00a0If a function contains the point <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image008.gif\" width=\"31\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0then the inverse of the function contains the point <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image010.gif\" width=\"31\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li><strong><em>Arithmetic series<\/em><\/strong> and <strong><em>geometric\u00a0series<\/em><\/strong> are expressions\u00a0for the sums of the terms in the sequence.<\/li>\n<li><strong><em>Linear equations<\/em><\/strong> have the form <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image012.gif\" width=\"85\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>, where <em><span style=\"text-decoration: none;\">m<\/span><\/em><span style=\"text-decoration: none;\"> = slope and <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\"> = <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercept. Any<\/span> point on the graph of a linear equation is a solution\u00a0to the equation.<\/li>\n<li>Solutions to <strong><em>linear\u00a0inequalities<\/em><\/strong> can occupy a region of the coordinate\u00a0plane that is bounded by a straight line. There can be many\u00a0solutions to linear inequalities.\u00a0Two lines that are <strong><em>parallel<\/em><\/strong> will\u00a0have the same slope, <span style=\"text-decoration: none;\">but\u00a0different <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-intercepts. <\/span><\/li>\n<li>Two lines that are <strong><em>perpendicular\u00a0<\/em><\/strong>will have slopes that are <span style=\"text-decoration: none;\"> negative reciprocals of each other<\/span>.<\/li>\n<li><strong><em>Systems of linear equations<\/em><\/strong> will\u00a0have a common <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p1_clip_image014.gif\" width=\"35\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/> point that is the solution to the system, if one exists. Not\u00a0every system will have a solution.<\/li>\n<\/ul>\n<section>\n<h3><strong>How are operations performed with polynomials?<\/strong><\/h3>\n<p>A polynomial is an expression that can be written as the sum of\u00a0the <abbr title=\"the individual elements of a polynomial that are added or subtracted\">terms<\/abbr> <img loading=\"lazy\" decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image002.gif\" width=\"20\" height=\"14\" name=\"graphics3\" border=\"0\" \/> where <em>b<\/em> is a positive integer.<\/p>\n<p>The exponent on each variable is the <abbr title=\"The value of the greatest exponent in a polynomial expression or equation. For example has a degree of 3.\">degree<\/abbr> of each term, and polynomials are usually written in <abbr title=\"A polynomial with the degree of each term written in descending order. \">standard\u00a0form<\/abbr>, meaning the terms are ordered in such a way\u00a0that the degrees are placed in descending order.<\/p>\n<p>For example,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image004.gif\" width=\"77\" height=\"14\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This is a three-term polynomial written in standard form. The\u00a0degree of the polynomial is the largest degree of its individual\u00a0terms, which is 3.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Anytime a polynomial is written in standard form, the degree of the\u00a0polynomial is just the exponent on the left-most term<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the degree of the polynomial <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image006.gif\" width=\"85\" height=\"14\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>?<\/p>\n<ol>\n<li>0<\/li>\n<li>1<\/li>\n<li>4<\/li>\n<li>6<\/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. Because the polynomial is written in\u00a0standard form, the degree is the exponent of the left-most term,\u00a0which in this case is 6.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which polynomial has a degree of 2, and is written in standard\u00a0form?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image008.gif\" width=\"54\" height=\"14\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image010.gif\" width=\"36\" height=\"14\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image012.gif\" width=\"34\" height=\"14\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image014.gif\" width=\"58\" height=\"14\" name=\"graphics9\" align=\"BOTTOM\" 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 B. All of the answers have a degree of\u00a02, but only B is written in standard form with the terms in\u00a0descending order by degree.<\/p>\n<\/div>\n<\/section>\n<p>While there are no limits to the degree ad complexity of\u00a0polynomials, we will focus on polynomials that have a degree of\u00a0four or less.<\/p>\n<ul>\n<li>Polynomials that have a degree\u00a0of one are called <em><span style=\"text-decoration: none;\"><em>linear<\/em><\/span><\/em> and are\u00a0represented graphically by a straight line. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image016.gif\" width=\"14\" height=\"11\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image018.gif\" width=\"45\" height=\"11\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/> are examples of linear polynomials.<\/li>\n<li>Polynomials that have a degree\u00a0of two are called quadratic and are represented graphically by a parabola. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image020.gif\" width=\"66\" height=\"14\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/> is an example of a quadratic equation.<\/li>\n<li>Polynomials with a degree of\u00a0three are called cubic. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image022.gif\" width=\"35\" height=\"14\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/> is an example\u00a0of a cubic polynomial.<\/li>\n<li>Polynomials with a degree of four are called quartic. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image024.gif\" width=\"64\" height=\"14\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/> is an example of a quartic equation.<\/li>\n<\/ul>\n<p>Polynomials can be further classified based on the number of\u00a0terms. A polynomial with one term is called a monomial, a polynomial with two terms is a binomial, and a polynomial with\u00a0three terms is a trinomial.<\/p>\n<p>The following table helps clarify the difference between the\u00a0classification by terms and degree.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"5\">\n<h4>Polynomial Classification<\/h4>\n<\/th>\n<\/tr>\n<tr>\n<th>Polynomial<\/th>\n<th>Terms<\/th>\n<th>Term Classification<\/th>\n<th>Degree<\/th>\n<th>Degree Classification<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image026.gif\" width=\"63\" height=\"14\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td>3<\/td>\n<td>Trinomial<\/td>\n<td>3<\/td>\n<td>Cubic<\/td>\n<\/tr>\n<tr>\n<td>2x + 1<\/td>\n<td>2<\/td>\n<td>Binomial<\/td>\n<td>1<\/td>\n<td>Linear<\/td>\n<\/tr>\n<tr>\n<td>-5<\/td>\n<td>1<\/td>\n<td>Monomial<\/td>\n<td>0<\/td>\n<td>Constant<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image028.gif\" width=\"62\" height=\"14\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td>3<\/td>\n<td>Trinomial<\/td>\n<td>2<\/td>\n<td>Quadratic<\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p2_clip_image030.gif\" width=\"33\" height=\"14\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td>2<\/td>\n<td>Binomial<\/td>\n<td>4<\/td>\n<td>Quartic<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>What are the different polynomial operations?<\/h3>\n<h3>Adding or Subtracting Polynomials<\/h3>\n<p>To add or subtract polynomials, the <abbr title=\" terms in an expression that have the same variable raised to the same exponent\">like terms<\/abbr> must be added or subtracted.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image002.gif\" width=\"332\" height=\"17\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>Multiplying Polynomials<\/h3>\n<p>Multiplying polynomials requires distributing each term in the\u00a0first polynomial to every term in the second polynomial, and then\u00a0simplifying like terms. The process is easier when one of the\u00a0polynomials is a monomial.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image004.gif\" width=\"238\" height=\"17\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Multiplying a binomial and a trinomial is slightly more\u00a0difficult.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image006.gif\" width=\"137\" height=\"14\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>To solve this problem, distribute each term in the first\u00a0polynomial (2x-1), to every term in the second polynomial, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image008.gif\" width=\"80\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image010.gif\" width=\"453\" height=\"14\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Continue simplifying by multiplying the terms.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image012.gif\" width=\"179\" height=\"11\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The final step is to combine like terms and write the\u00a0polynomial in standard form.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image014.gif\" width=\"112\" height=\"11\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Keep in mind the rule for exponents when multiplying\u00a0like variables add the exponents together.<\/p>\n<\/div>\n<p>Multiplying two binomials is very common in algebra. FOIL is an\u00a0acronym that can help you remember the steps in this\u00a0multiplication process.<\/p>\n<p>The letters in FOIL stand for &#8220;<strong><span style=\"text-decoration: none;\">f<\/span><\/strong><span style=\"text-decoration: none;\">irst, <\/span><strong><span style=\"text-decoration: none;\">o<\/span><\/strong><span style=\"text-decoration: none;\">utside, <\/span><strong><span style=\"text-decoration: none;\">i<\/span><\/strong><span style=\"text-decoration: none;\">nside,\u00a0and <\/span><strong><span style=\"text-decoration: none;\">l<\/span><\/strong><span style=\"text-decoration: none;\">ast.&#8221;\u00a0To multiply two binomials, multiply the <\/span><strong><span style=\"text-decoration: none;\">f<\/span><\/strong><span style=\"text-decoration: none;\">irst, <\/span><strong><span style=\"text-decoration: none;\">o<\/span><\/strong><span style=\"text-decoration: none;\">utside, <\/span><strong><span style=\"text-decoration: none;\">i<\/span><\/strong><span style=\"text-decoration: none;\">nside,\u00a0and <\/span><strong><span style=\"text-decoration: none;\">l<\/span><\/strong><span style=\"text-decoration: none;\">ast\u00a0term<\/span>s of the polynomials, in that order.<\/p>\n<p>First, write out the two binomials to be multiplied, placing\u00a0them next to each other.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image016.gif\" width=\"90\" height=\"18\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The <strong><span style=\"text-decoration: none;\">f<\/span><\/strong>irst\u00a0two terms in each bino<span style=\"text-decoration: none;\">mial are\u00a02<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> and <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> . Multiplying them results in 2<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">\u00b2\u00a0. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">The <\/span><strong><span style=\"text-decoration: none;\">o<\/span><\/strong><span style=\"text-decoration: none;\">utside\u00a0two terms are 2<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> and 1. Multiplying them results in 2<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> . <\/span><\/p>\n<p><span style=\"text-decoration: none;\">The <\/span><strong><span style=\"text-decoration: none;\">i<\/span><\/strong><span style=\"text-decoration: none;\">nside\u00a0two terms are 5 and <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> . Multiplying them results in 5<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><\/p>\n<p><span style=\"text-decoration: none;\">The <\/span><strong><span style=\"text-decoration: none;\">l<\/span><\/strong><span style=\"text-decoration: none;\">ast\u00a0two terms are 5 and 1. Multiplying them results in 5. <\/span><\/p>\n<p style=\"text-decoration: none;\">Next, add all terms that resulted\u00a0from the FOIL multiplication.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s1_p3_html_m3680af2b.gif\" width=\"115\" height=\"21\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>To simplify into the final answer, combine all like terms.\u00a0Remember that like terms have to have the same exponent on the\u00a0variable.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image020.gif\" width=\"94\" height=\"11\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3><strong>Polynomial Division<\/strong><\/h3>\n<p>Like multiplication, polynomials can be divided easily if there\u00a0is a monomial involved.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image022.gif\" width=\"83\" height=\"37\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Because the divisor is a monomial, each term in the numerator\u00a0can be separated and divided by 2<em>x<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image024.gif\" width=\"197\" height=\"37\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Simplify each ratio to find the final answer.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image026.gif\" width=\"70\" height=\"34\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now try an example of dividing by a binomial.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image028.gif\" width=\"61\" height=\"37\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This is still a relatively easy problem, because the numerator\u00a0can be factored into two binomials, canceling out the denominator.\u00a0(The next section will address factoring.)<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image030.gif\" width=\"206\" height=\"40\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Sometimes factoring and canceling cannot be used to solve a\u00a0problem.<\/p>\n<p>For example:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image032.gif\" width=\"67\" height=\"37\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The quadratic does not factor with integer roots. Therefore,\u00a0divide using long division.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image034.gif\" width=\"101\" height=\"143\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The final answer is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image036.gif\" width=\"73\" height=\"34\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<\/p>\n<p>Check the answer by multiplying each term by <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image038.gif\" width=\"36\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/> and working backward toward\u00a0the original polynomial.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s1_p3_clip_image040.gif\" width=\"128\" height=\"59\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Because the new answer matches the original polynomial, the\u00a0answer is correct.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><!--<a href=\"math_02_09.html\" class=\"button button-primary\">\u2b05 Previous Lesson<\/a>--><br \/>\n<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\/factoring-polynomials\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Workshop Index\u00a0Next\u00a0Lesson \u27a1 Algebra &amp; Functions Objective In this lesson, you will study more advanced algebra topics such as rational polynomials, factoring, quadratics,\u00a0conics, exponents, and the Fundamental Theorem of Algebra. Previously Covered: Functions are based on rules that\u00a0define inputs and outputs. For each\u00a0input, there is exactly one output. Composite functions are evaluated in\u00a0series, meaning that [&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-94","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/94","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=94"}],"version-history":[{"count":13,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/94\/revisions"}],"predecessor-version":[{"id":790,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/94\/revisions\/790"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}