{"id":2110,"date":"2015-11-12T18:30:41","date_gmt":"2015-11-12T18:30:41","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=2110"},"modified":"2015-11-12T18:30:41","modified_gmt":"2015-11-12T18:30:41","slug":"using-the-binomial-theorem-to-find-a-single-term","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-the-binomial-theorem-to-find-a-single-term\/","title":{"raw":"Using the Binomial Theorem to Find a Single Term","rendered":"Using the Binomial Theorem to Find a Single Term"},"content":{"raw":"

Expanding a binomial with a high exponent such as [latex]{\\left(x+2y\\right)}^{16}[\/latex] can be a lengthy process.\n\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.\n\nNote the pattern of coefficients in the expansion of [latex]{\\left(x+y\\right)}^{5}[\/latex].\n<\/p>

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

A General Note: The (r+1)th Term of a Binomial Expansion<\/h3>\nThe [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion of [latex]{\\left(x+y\\right)}^{n}[\/latex] is:\n
[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<\/div>\n
\n

How To: Given a binomial, write a specific term without fully expanding.<\/h3>\n
  1. Determine the value of [latex]n[\/latex] according to the exponent.<\/li>\n\t
  2. Determine [latex]\\left(r+1\\right)[\/latex].<\/li>\n\t
  3. Determine [latex]r[\/latex].<\/li>\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>\n<\/ol><\/div>\n
    \n

    Example 3: Writing a Given Term of a Binomial Expansion<\/h3>\nFind the tenth term of [latex]{\\left(x+2y\\right)}^{16}[\/latex] without fully expanding the binomial.\n\n<\/div>\n
    \n

    Solution<\/h3>\nBecause we are looking for the tenth term, [latex]r+1=10[\/latex], we will use [latex]r=9[\/latex] in our calculations.\n
    [latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n
    [latex]\\left(\\begin{array}{c}16\\\\ 9\\end{array}\\right){x}^{16 - 9}{\\left(2y\\right)}^{9}=5\\text{,}857\\text{,}280{x}^{7}{y}^{9}[\/latex]<\/div>\n<\/div>\n
    \n

    Try It 3<\/h3>\nFind the sixth term of [latex]{\\left(3x-y\\right)}^{9}[\/latex] without fully expanding the binomial.\n\nSolution<\/a>\n\n<\/div>","rendered":"

    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].\n<\/p>\n

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

    The second term is [latex]\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}y[\/latex]. The third term is [latex]\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}[\/latex]. We can generalize this result.<\/p>\n

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

    A General Note: The (r+1)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{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<\/div>\n
    \n

    How To: Given a binomial, write a specific term without fully expanding.<\/h3>\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<\/div>\n
      \n

      Example 3: Writing a Given Term of a Binomial Expansion<\/h3>\n

      Find the tenth term of [latex]{\\left(x+2y\\right)}^{16}[\/latex] without fully expanding the binomial.<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      Because we are looking for the tenth term, [latex]r+1=10[\/latex], we will use [latex]r=9[\/latex] in our calculations.<\/p>\n

      [latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n
      [latex]\\left(\\begin{array}{c}16\\\\ 9\\end{array}\\right){x}^{16 - 9}{\\left(2y\\right)}^{9}=5\\text{,}857\\text{,}280{x}^{7}{y}^{9}[\/latex]<\/div>\n<\/div>\n
      \n

      Try It 3<\/h3>\n

      Find the sixth term of [latex]{\\left(3x-y\\right)}^{9}[\/latex] without fully expanding the binomial.<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
      \n\t\t\t\t\t\t\t
      CC licensed content, Specific attribution<\/div>