{"id":982,"date":"2025-07-17T22:29:49","date_gmt":"2025-07-17T22:29:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=982"},"modified":"2026-03-12T17:24:43","modified_gmt":"2026-03-12T17:24:43","slug":"zeros-of-polynomial-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/zeros-of-polynomial-functions-learn-it-4\/","title":{"raw":"Zeros of Polynomial Functions: Learn It 4","rendered":"Zeros of Polynomial Functions: Learn It 4"},"content":{"raw":"

Linear Factorization Theorem and Complex Conjugate Theorem<\/h2>\r\nA vital implication of the Fundamental Theorem of Algebra is that a polynomial function of degree [latex]n[\/latex] will have [latex]n[\/latex] zeros in the set of complex numbers if we allow for multiplicities. This means that we can factor the polynomial function into [latex]n[\/latex] factors.\r\n\r\n
\r\n
\r\n

Linear Factorization Theorem<\/h3>\r\nThe Linear Factorization Theorem<\/strong> tells us that a polynomial function will have the same number of factors as its degree, and each factor will be of the form [latex](x \u2013 c)[\/latex] where [latex]c[\/latex] is a complex number.\r\n\r\n<\/div>\r\n<\/section>\r\n
[ohm_question hide_question_numbers=1]318901[\/ohm_question]\u00a0<\/span><\/section><\/div>\r\n
When you learned to divide complex numbers, you multiplied the top and bottom of the quotient of complex numbers deliberately by the conjugate of the denominator so that the imaginary part would eliminate from the denominator. That is, multiplying complex conjugates eliminates the imaginary part.<\/section>Let [latex]f[\/latex]\u00a0be a polynomial function with real coefficients and suppose [latex]a+bi\\text{, }b\\ne 0[\/latex],\u00a0is a zero of [latex]f\\left(x\\right)[\/latex].\u00a0Then, by the Factor Theorem, [latex]x-\\left(a+bi\\right)[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex].\r\n\r\nFor [latex]f[\/latex]\u00a0to have real coefficients, [latex]x-\\left(a-bi\\right)[\/latex]\u00a0must also be a factor of [latex]f\\left(x\\right)[\/latex].\u00a0This is true because any factor other than [latex]x-\\left(a-bi\\right)[\/latex],\u00a0when multiplied by [latex]x-\\left(a+bi\\right)[\/latex],\u00a0will leave imaginary components in the product.\r\n\r\nOnly multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients.\r\n\r\nIn other words, if a polynomial function [latex]f[\/latex]\u00a0with real coefficients has a complex zero [latex]a+bi[\/latex],\u00a0then the complex conjugate [latex]a-bi[\/latex]\u00a0must also be a zero of [latex]f\\left(x\\right)[\/latex]. This is called the Complex Conjugate Theorem<\/strong>.\r\n\r\n
\r\n

Complex Conjugate Theorem<\/h3>\r\nIf the polynomial function [latex]f[\/latex]\u00a0has real coefficients and a complex zero of the form [latex]a+bi[\/latex],\u00a0then the complex conjugate of the zero, [latex]a-bi[\/latex],\u00a0is also a zero.\r\n\r\n<\/section>
How To: Given the zeros of a polynomial function [latex]f[\/latex] and a point [latex]\\left(c\\text{, }f(c)\\right)[\/latex]\u00a0on the graph of [latex]f[\/latex], use the Linear Factorization Theorem to find the polynomial function<\/strong>\r\n
    \r\n \t
  1. Use the zeros to construct the linear factors of the polynomial.<\/li>\r\n \t
  2. Multiply the linear factors to expand the polynomial.<\/li>\r\n \t
  3. Substitute [latex]\\left(c,f\\left(c\\right)\\right)[\/latex] into the function to determine the leading coefficient.<\/li>\r\n \t
  4. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Find a fourth degree polynomial with real coefficients that has zeros of [latex]\u20133[\/latex], [latex]2[\/latex], [latex]i[\/latex], such that [latex]f\\left(-2\\right)=100[\/latex].[reveal-answer q=\"412896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"412896\"]Because [latex]x=i[\/latex]\u00a0is a zero, by the Complex Conjugate Theorem [latex]x=-i[\/latex]\u00a0is also a zero. The polynomial must have factors of [latex]\\left(x+3\\right),\\left(x - 2\\right),\\left(x-i\\right)[\/latex], and [latex]\\left(x+i\\right)[\/latex]. Since we are looking for a degree [latex]4[\/latex] polynomial and now have four zeros, we have all four factors. Let\u2019s begin by multiplying these factors.\r\n

    [latex]\\begin{array}{l}f\\left(x\\right)=a\\left(x+3\\right)\\left(x - 2\\right)\\left(x-i\\right)\\left(x+i\\right)\\\\ f\\left(x\\right)=a\\left({x}^{2}+x - 6\\right)\\left({x}^{2}+1\\right)\\\\ f\\left(x\\right)=a\\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\\right)\\end{array}[\/latex]<\/p>\r\nWe need to find [latex]a[\/latex] to ensure [latex]f\\left(-2\\right)=100[\/latex]. Substitute [latex]x=-2[\/latex] and [latex]f\\left(2\\right)=100[\/latex]\r\ninto [latex]f\\left(x\\right)[\/latex].\r\n

    [latex]\\begin{array}{l}100=a\\left({\\left(-2\\right)}^{4}+{\\left(-2\\right)}^{3}-5{\\left(-2\\right)}^{2}+\\left(-2\\right)-6\\right)\\hfill \\\\ 100=a\\left(-20\\right)\\hfill \\\\ -5=a\\hfill \\end{array}[\/latex]<\/p>\r\nSo the polynomial function is:\r\n

    [latex]f\\left(x\\right)=-5\\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\\right)[\/latex]<\/p>\r\n

    or<\/p>\r\n

    [latex]f\\left(x\\right)=-5{x}^{4}-5{x}^{3}+25{x}^{2}-5x+30[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe found that both [latex]i[\/latex] and [latex]\u2013i[\/latex] were zeros, but only one of these zeros needed to be given. If [latex]i[\/latex]\u00a0is a zero of a polynomial with real coefficients, then [latex]\u2013i[\/latex]\u00a0must also be a zero of the polynomial because [latex]\u2013i[\/latex]\u00a0is the complex conjugate of [latex]i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321187[\/ohm_question]<\/section>","rendered":"

    Linear Factorization Theorem and Complex Conjugate Theorem<\/h2>\n

    A vital implication of the Fundamental Theorem of Algebra is that a polynomial function of degree [latex]n[\/latex] will have [latex]n[\/latex] zeros in the set of complex numbers if we allow for multiplicities. This means that we can factor the polynomial function into [latex]n[\/latex] factors.<\/p>\n

    \n
    \n

    Linear Factorization Theorem<\/h3>\n

    The Linear Factorization Theorem<\/strong> tells us that a polynomial function will have the same number of factors as its degree, and each factor will be of the form [latex](x \u2013 c)[\/latex] where [latex]c[\/latex] is a complex number.<\/p>\n<\/div>\n<\/section>\n

    \n