{"id":1752,"date":"2025-07-28T18:59:41","date_gmt":"2025-07-28T18:59:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1752"},"modified":"2026-03-26T19:23:34","modified_gmt":"2026-03-26T19:23:34","slug":"binomial-theorem-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/binomial-theorem-learn-it-3\/","title":{"raw":"Binomial Theorem: Learn It 3","rendered":"Binomial Theorem: Learn It 3"},"content":{"raw":"

Using the Binomial Theorem to Find a Single Term<\/h2>\r\nExpanding a binomial with a high exponent such as [latex]{\\left(x+2y\\right)}^{16}[\/latex] can be a lengthy process.\r\n\r\nSometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.\r\n\r\nNote the pattern of coefficients in the expansion of [latex]{\\left(x+y\\right)}^{5}[\/latex].\r\n

[latex]{\\left(x+y\\right)}^{5}={x}^{5}+\\left(\\begin{gathered}5\\\\ 1\\end{gathered}\\right){x}^{4}y+\\left(\\begin{gathered}5\\\\ 2\\end{gathered}\\right){x}^{3}{y}^{2}+\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right){x}^{2}{y}^{3}+\\left(\\begin{gathered}5\\\\ 4\\end{gathered}\\right)x{y}^{4}+{y}^{5}[\/latex]<\/p>\r\nThe second term is [latex]\\left(\\begin{gathered}5\\\\ 1\\end{gathered}\\right){x}^{4}y[\/latex]. The third term is [latex]\\left(\\begin{gathered}5\\\\ 2\\end{gathered}\\right){x}^{3}{y}^{2}[\/latex].\r\n\r\nWe can generalize this result.\r\n

[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right){x}^{n-r}{y}^{r}[\/latex]<\/p>\r\n\r\n

\r\n

the [latex](r+1)[\/latex]th term of a binomial expansion<\/h3>\r\nThe [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion of [latex]{\\left(x+y\\right)}^{n}[\/latex] is:\r\n

[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right){x}^{n-r}{y}^{r}[\/latex]<\/p>\r\n\r\n<\/section>

How To: Given a binomial, write a specific term without fully expanding.<\/strong>\r\n
    \r\n \t
  1. Determine the value of [latex]n[\/latex] according to the exponent.<\/li>\r\n \t
  2. Determine [latex]\\left(r+1\\right)[\/latex].<\/li>\r\n \t
  3. Determine [latex]r[\/latex].<\/li>\r\n \t
  4. Replace [latex]r[\/latex] in the formula for the [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion.<\/li>\r\n<\/ol>\r\n<\/section>
    Find the tenth term of [latex]{\\left(x+2y\\right)}^{16}[\/latex] without fully expanding the binomial.[reveal-answer q=\"495201\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495201\"]Because we are looking for the tenth term, [latex]r+1=10[\/latex], we will use [latex]r=9[\/latex] in our calculations.\r\n

    [latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right){x}^{n-r}{y}^{r}[\/latex]<\/p>\r\n

    [latex]\\left(\\begin{gathered}16\\\\ 9\\end{gathered}\\right){x}^{16 - 9}{\\left(2y\\right)}^{9}=5\\text{,}857\\text{,}280{x}^{7}{y}^{9}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]322006[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]322007[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]322008[\/ohm_question]<\/section>","rendered":"

    Using the Binomial Theorem to Find a Single Term<\/h2>\n

    Expanding a binomial with a high exponent such as [latex]{\\left(x+2y\\right)}^{16}[\/latex] can be a lengthy process.<\/p>\n

    Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.<\/p>\n

    Note the pattern of coefficients in the expansion of [latex]{\\left(x+y\\right)}^{5}[\/latex].<\/p>\n

    [latex]{\\left(x+y\\right)}^{5}={x}^{5}+\\left(\\begin{gathered}5\\\\ 1\\end{gathered}\\right){x}^{4}y+\\left(\\begin{gathered}5\\\\ 2\\end{gathered}\\right){x}^{3}{y}^{2}+\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right){x}^{2}{y}^{3}+\\left(\\begin{gathered}5\\\\ 4\\end{gathered}\\right)x{y}^{4}+{y}^{5}[\/latex]<\/p>\n

    The second term is [latex]\\left(\\begin{gathered}5\\\\ 1\\end{gathered}\\right){x}^{4}y[\/latex]. The third term is [latex]\\left(\\begin{gathered}5\\\\ 2\\end{gathered}\\right){x}^{3}{y}^{2}[\/latex].<\/p>\n

    We can generalize this result.<\/p>\n

    [latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right){x}^{n-r}{y}^{r}[\/latex]<\/p>\n

    \n

    the [latex](r+1)[\/latex]th term of a binomial expansion<\/h3>\n

    The [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion of [latex]{\\left(x+y\\right)}^{n}[\/latex] is:<\/p>\n

    [latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right){x}^{n-r}{y}^{r}[\/latex]<\/p>\n<\/section>\n

    How To: Given a binomial, write a specific term without fully expanding.<\/strong><\/p>\n
      \n
    1. Determine the value of [latex]n[\/latex] according to the exponent.<\/li>\n
    2. Determine [latex]\\left(r+1\\right)[\/latex].<\/li>\n
    3. Determine [latex]r[\/latex].<\/li>\n
    4. Replace [latex]r[\/latex] in the formula for the [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion.<\/li>\n<\/ol>\n<\/section>\n
      Find the tenth term of [latex]{\\left(x+2y\\right)}^{16}[\/latex] without fully expanding the binomial.<\/p>\n