{"id":2007,"date":"2019-08-26T12:15:12","date_gmt":"2019-08-26T17:15:12","guid":{"rendered":"http:\/\/www.kaptest.com\/blog\/prep\/?p=2007"},"modified":"2023-08-29T21:10:01","modified_gmt":"2023-08-29T21:10:01","slug":"psat-math-polynomials","status":"publish","type":"post","link":"https:\/\/wpapp.kaptest.com\/study\/psat\/psat-math-polynomials\/","title":{"rendered":"PSAT Math: Polynomials"},"content":{"rendered":"<p>By now you\u2019re used to seeing equations, exponents, and variables; another important topic you are sure to see on the\u00a0<span class=\"sat-exclude\">PSAT<\/span>\u00a0is polynomials. A\u00a0<b>polynomial<\/b>\u00a0is an expression comprised of variables, exponents, and coefficients, and the only operations involved are addition, subtraction, multiplication, division (by constants\u00a0<i>only<\/i>), and non-negative integer exponents. A polynomial can have one or multiple terms.<\/p>\n<article  class=\"iconbox iconbox_left_content    avia-builder-el-0  el_before_av_heading  avia-builder-el-first  \"  itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/BlogPosting\" itemprop=\"blogPost\" ><div class=\"iconbox_icon heading-color\" aria-hidden='true' data-av_icon='\ue812' data-av_iconfont='entypo-fontello'  ><\/div><div class=\"iconbox_content\"><header class=\"entry-content-header\"><h3 class='iconbox_content_title  '  itemprop=\"headline\"  >Note<\/h3><\/header><div class='iconbox_content_container  '  itemprop=\"text\"  ><p>Remember that a constant, such as 47, is considered a polynomial; this is the same as<span class=\"no-break\">\u00a047<i>x<\/i><sup>\u200a0<\/sup>.<\/span>\u00a0Also, keep in mind that for an expression to be a polynomial, division by a constant is allowed, but division by a variable is not.<\/p>\n<\/div><\/div><footer class=\"entry-footer\"><\/footer><\/article>\n<p>Identifying <b>like terms<\/b> is an important skill that will serve you well on Test Day. To simplify polynomial expressions, you combine like terms just as you did with linear expressions and equations (<i>x<\/i> terms with <i>x<\/i> terms, constants with constants). To have like terms, the types of variables present and their exponents must match. For example, 2<i>xy<\/i> and \u22124<i>xy<\/i> are like terms; <i>x<\/i> and <i>y<\/i> are present in both, and their corresponding exponents are identical. However, 2<i>x<\/i><sup>\u200a\u200a<\/sup><sup>2<\/sup><i>y<\/i> and 3<i>xy<\/i> are not like terms because the exponents on <i>x<\/i> do not match. A few more examples follow:<\/p>\n<table cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td valign=\"middle\">Like terms<\/td>\n<td valign=\"middle\">7<i>x<\/i>, 3<i>x<\/i>, 5<i>x<\/i><\/td>\n<td valign=\"middle\">3, 15, 900<\/td>\n<td valign=\"middle\"><i>xy<\/i><sup>2<\/sup>, 7<i>xy<\/i><sup>2<\/sup>, \u20132<i>xy<\/i><sup>2<\/sup><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>Not<\/i> like terms<\/td>\n<td valign=\"middle\">3, <i>x<\/i>, <i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup><\/td>\n<td valign=\"middle\">4<i>x<\/i>, 4<i>y<\/i>, 4z<\/td>\n<td valign=\"middle\"><i>xy<\/i><sup>\u200a<\/sup><sup>2<\/sup>, <i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup><i><sup>y<\/sup><\/i>, 2<i>xy<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>You can also\u00a0<b>evaluate<\/b>\u00a0a polynomial expression (just like any other expression) for given values in its domain. For example, suppose you&#8217;re given the polynomial expression <i>x<\/i><sup>\u200a<\/sup><sup>3<\/sup> + 5<i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup> + 1. At <i>x<\/i> = \u20131, the value of the expression is (\u20131)<sup>3<\/sup> + 5(\u20131)<sup>2<\/sup> + 1, which simplifies to \u20131 + 5 + 1 = 5.<\/p>\n<p>A polynomial can be named based on its <b>degree<\/b>. For a single-variable polynomial, the degree is the highest power on the variable. For example, the degree of 3<i>x<\/i><sup>\u200a<\/sup><sup>4<\/sup> \u2013 2<i>x<\/i><sup>\u200a<\/sup><sup>3<\/sup> + <i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup> \u2013 5<i>x<\/i> + 2 is 4 because the highest power of <i>x<\/i> is 4. For a multi-variable polynomial, the degree is the highest sum of the exponents on any one term. For example, the degree of 3<i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup><i>y<\/i><sup>2<\/sup> \u2013 5<i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup><i>y<\/i> + <i>x<\/i><sup>\u200a<\/sup><sup>3<\/sup> is 4 because the sum of the exponents in the term 3<i>x<\/i><sup>\u200a<\/sup><sup>2<\/sup><i>y<\/i><sup>2\u00a0<\/sup>equals 4.<\/p>\n<p><div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h5    avia-builder-el-1  el_after_av_icon_box  el_before_av_image  '><h5 class='av-special-heading-tag '  itemprop=\"headline\"  >Zeros and Roots<\/h5><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nOn Test Day you might be asked about the nature of the <b>zeros<\/b> or <b>roots<\/b> of a polynomial. Simply put, zeros are the <i>x<\/i>-intercepts of a polynomial\u2019s graph, which can be found by setting each factor of the polynomial equal to 0.<\/p>\n<p>For example, in the polynomial equation <i>y<\/i> = (<i>x<\/i> + 6)(<i>x<\/i> \u2013 2)<sup>2<\/sup>, you would have three equations: <i>x<\/i> + 6 = 0, <i>x<\/i> \u2013 2 = 0, and <i>x<\/i> \u2013 2 = 0 (because <i>x<\/i> \u2013 2 is squared, that binomial appears twice in the equation). Solving for <i>x<\/i> in each yields \u20136, 2, and 2; we say that the equation has two zeros: \u20136 and 2.<\/p>\n<p>Zeros can have varying levels of <b>multiplicity<\/b>, which is the number of times that a factor appears in the polynomial equation. In the preceding example, <i>x<\/i> + 6 appears once in the equation, so its corresponding zero (\u20136) is called a <b>simple zero<\/b>. Because <i>x<\/i> \u2013 2 appears twice in the equation, its corresponding zero (2) is called a <b>double zero<\/b>.<\/p>\n<p>You can recognize the multiplicity of a zero from the polynomial\u2019s graph as well. Following is the graph of <i>y<\/i> = (<i>x<\/i> + 6)(<i>x<\/i> \u2013 2)<sup>2<\/sup>.<\/p>\n<div  class='avia-image-container avia_animated_image avia_animate_when_almost_visible top-to-bottom av-styling-    avia-builder-el-2  el_after_av_heading  el_before_av_heading  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/c08_cb_01-300x215.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div>\n<p>When a polynomial has a simple zero (multiplicity 1) or any zero with an odd multiplicity, its graph will cross the <i>x<\/i>-axis (as it does at <i>x<\/i> = \u20136 in the graph above). When a polynomial has a double zero (multiplicity 2) or any zero with an even multiplicity, it just touches the <i>x<\/i>-axis (as it does at\u00a0<i>x<\/i> = 2 in the graph above).<\/p>\n<p><div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-3  el_after_av_image  el_before_av_promobox  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >PSAT Math Practice Question: Polynomials<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\n\t<div  style='background:#ffffff;color:#444444;border-color:#444444;' class='av_promobox  avia-button-no   avia-builder-el-4  el_after_av_heading  el_before_av_hr '>\t\t<div class='avia-promocontent'><p>\n1. If <i>A<\/i> and <i>B<\/i> are polynomial expressions such that <i>A<\/i> = 24<i>xy<\/i> + 13 and <i>B<\/i> = 8<i>xy<\/i> + 1, how much greater is <i>A<\/i> than <i>B\u00a0<\/i>?<\/p>\n<p>A. 32<i>xy<\/i> + 14<br \/>\nB. 16<i>xy<\/i> + 14<br \/>\nC. 16<i>xy<\/i> + 12<br \/>\nD. 32<i>xy<\/i> + 13<\/p>\n<\/div><\/div><\/p>\n<div  style='height:10px' class='hr hr-invisible   avia-builder-el-5  el_after_av_promobox  avia-builder-el-last '><span class='hr-inner ' ><span class='hr-inner-style'><\/span><\/span><\/div>\n<p>Use the Kaplan Method for Math to solve this question, working through it step-by-step. The following table shows Kaplan\u2019s strategic thinking on the left, along with suggested math scratchwork on the right.<\/p>\n<table cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td valign=\"middle\"><b>Strategic Thinking<\/b><\/td>\n<td valign=\"middle\"><b>Math Scratchwork<\/b><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 1: Read the question, identifying and organizing important information as you go<\/b><br \/>\nDon\u2019t let the unusual wording fool you. To find how much greater <i>A<\/i> is, just do what you would do for two numbers: Subtract the smaller from the larger.<\/td>\n<td valign=\"middle\"><i>A \u2013 B<\/i><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 2: Choose the best strategy to answer the question<\/b><br \/>\n<i>What\u2019s your first step?<\/i><\/p>\n<p>Substitute the correct expressions for <i>A<\/i> and <i>B<\/i>. Distribute the \u20131 outside the second set of parentheses. Be careful here; this is an easy place to make a mistake.<\/p>\n<p><i>And afterward?<\/i><\/p>\n<p>Combine like terms. Rearranging so that like terms are next to each other helps here.<\/td>\n<td valign=\"middle\">(24<i>xy<\/i> + 13) \u2013 (8<i>xy<\/i> + 1) =<\/p>\n<p>24<i>xy<\/i> + 13 \u2013 8<i>xy<\/i> \u2013 1=<\/p>\n<p>24<i>xy<\/i> \u2013 8<i>xy<\/i> + 13 \u2013 1=<\/p>\n<p>16<i>xy<\/i> + 12<\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 3: Check that you answered the\u00a0<i>right\u00a0<\/i>question<\/b><\/p>\n<p>No further simplification is possible; the correct answer is (C), so you\u2019re done.<\/td>\n<td valign=\"middle\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>By now you\u2019re used to seeing equations, exponents, and variables; another important topic you are sure to see on the\u00a0PSAT\u00a0is polynomials. A\u00a0polynomial\u00a0is an expression comprised of variables, exponents, and coefficients, and the only operations involved are addition, subtraction, multiplication, division (by constants\u00a0only), and non-negative integer exponents. A polynomial can have one or multiple terms. Identifying [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28734,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[240],"tags":[],"_links":{"self":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/2007"}],"collection":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/comments?post=2007"}],"version-history":[{"count":13,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/2007\/revisions"}],"predecessor-version":[{"id":44191,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/2007\/revisions\/44191"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media\/28734"}],"wp:attachment":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media?parent=2007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/categories?post=2007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/tags?post=2007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}