{"id":1976,"date":"2024-06-26T22:29:15","date_gmt":"2024-06-26T22:29:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1976"},"modified":"2026-02-09T20:40:53","modified_gmt":"2026-02-09T20:40:53","slug":"mastering-polynomial-functions-theorems-zeros-and-applications-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/mastering-polynomial-functions-theorems-zeros-and-applications-learn-it-6\/","title":{"raw":"Zeros of Polynomial Functions: Learn It 6","rendered":"Zeros of Polynomial Functions: Learn It 6"},"content":{"raw":"

Descartes\u2019 Rule of Signs<\/h2>\r\nThere is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex]f\\left(x\\right)[\/latex] and the number of positive real zeros.\r\n\r\nThere is a similar relationship between the number of sign changes in [latex]f\\left(-x\\right)[\/latex] and the number of negative real zeros.\r\n\r\n
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Descartes\u2019 Rule of Signs<\/h3>\r\nAccording to Descartes\u2019 Rule of Signs<\/strong>, if we let [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]\u00a0be a polynomial function with real coefficients:\r\n
    \r\n \t
  • The number of positive real zeros is either equal to the number of sign changes of the coefficients of [latex]f\\left(x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n \t
  • The number of negative real zeros is either equal to the number of sign changes of the coefficients of [latex]f\\left(-x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>
    Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for:\r\n
    [latex]f\\left(x\\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12[\/latex]<\/center>\r\n[reveal-answer q=\"143065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143065\"]Begin by determining the number of sign changes.\r\n\r\n[caption id=\"attachment_11813\" align=\"aligncenter\" width=\"534\"]\"Screen Analyzing f(x) using Descartes\u2019 Rule of Signs[\/caption]\r\n\r\nThere are two sign changes, so there are either [latex]2[\/latex] or [latex]0[\/latex] positive real roots. Next, we examine [latex]f\\left(-x\\right)[\/latex] to determine the number of negative real roots.\r\n\r\n[caption id=\"attachment_11814\" align=\"aligncenter\" width=\"536\"]\"Screen Analyzing f(x) using Descartes\u2019 Rule of Signs[\/caption]\r\n

    [latex]\\begin{array}{l}f\\left(-x\\right)=-{\\left(-x\\right)}^{4}-3{\\left(-x\\right)}^{3}+6{\\left(-x\\right)}^{2}-4\\left(-x\\right)-12\\hfill \\\\ f\\left(-x\\right)=-{x}^{4}+3{x}^{3}+6{x}^{2}+4x - 12\\hfill \\end{array}[\/latex]<\/p>\r\nAgain, there are two sign changes, so there are either [latex]2[\/latex] or [latex]0[\/latex] negative real roots.\r\n\r\nThere are four possibilities, as we can see below.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Positive Real Zeros<\/th>\r\nNegative Real Zeros<\/th>\r\nComplex Zeros<\/th>\r\nTotal Zeros<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe can confirm the numbers of positive and negative real roots by examining a graph of the function.\u00a0We can see from the graph that the function has 0 positive real roots and 2 negative real roots.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]24636[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]24637[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]24638[\/ohm2_question]<\/section>","rendered":"

    Descartes\u2019 Rule of Signs<\/h2>\n

    There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex]f\\left(x\\right)[\/latex] and the number of positive real zeros.<\/p>\n

    There is a similar relationship between the number of sign changes in [latex]f\\left(-x\\right)[\/latex] and the number of negative real zeros.<\/p>\n

    \n
    \n

    Descartes\u2019 Rule of Signs<\/h3>\n

    According to Descartes\u2019 Rule of Signs<\/strong>, if we let [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]\u00a0be a polynomial function with real coefficients:<\/p>\n

      \n
    • The number of positive real zeros is either equal to the number of sign changes of the coefficients of [latex]f\\left(x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n
    • The number of negative real zeros is either equal to the number of sign changes of the coefficients of [latex]f\\left(-x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n
      Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for:<\/p>\n
      [latex]f\\left(x\\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12[\/latex]<\/div>\n