{"id":821,"date":"2024-04-29T21:05:42","date_gmt":"2024-04-29T21:05:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=821"},"modified":"2025-08-21T23:05:24","modified_gmt":"2025-08-21T23:05:24","slug":"polynomial-basics-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-basics-learn-it-3\/","title":{"raw":"Polynomial Basics: Learn It 3","rendered":"Polynomial Basics: Learn It 3"},"content":{"raw":"

Multiplying Polynomials<\/h2>\r\nMultiplying polynomials is a step up from adding and subtracting them, but once you get the hang of it, it's pretty straightforward!\r\n\r\nTo multiply polynomials, we use what's called the distributive property<\/strong>. This means we take each term from the first polynomial and multiply it by every term in the second polynomial. After that, we just combine any like terms we find.\r\n\r\n
Distributive Property: [latex]a\\cdot(b+c) = a\\cdot b+a\\cdot c[\/latex]<\/section>
How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/strong>\r\n
    \r\n \t
  1. Multiply each term of the first polynomial by each term of the second.<\/li>\r\n \t
  2. Combine like terms.<\/li>\r\n \t
  3. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Find the product and simplify:[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]Solution\u00a0<\/strong>[latex]\\begin{align*} (2x+1)(3x^2-x+4) & = 2x(3x^2-x+4) + 1(3x^2-x+4) & \\text{Use the distributive property} \\\\ & = (6x^3-2x^2+8x) + (3x^2-x+4) & \\text{Multiply each term} \\\\ & = 6x^3 + (-2x^2+3x^2) + (8x-x) + 4 & \\text{Combine like terms} \\\\ & = 6x^3 + x^2 + 7x + 4 & \\text{Simplify to final form} \\end{align*}[\/latex]\r\n\r\n[reveal-answer q=\"122501\"]Analysis of multiplication using a table[\/reveal-answer]\r\n[hidden-answer a=\"122501\"]We can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\r\n\r\n\r\n\r\n\r\n\r\n
    <\/td>\r\n[latex]3{x}^{2}[\/latex]<\/strong><\/td>\r\n[latex]-x[\/latex]<\/strong><\/td>\r\n[latex]+4[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
    [latex]2x[\/latex]<\/strong><\/td>\r\n[latex]6{x}^{3}\\\\[\/latex]<\/td>\r\n[latex]-2{x}^{2}[\/latex]<\/td>\r\n[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]+1[\/latex]<\/strong><\/td>\r\n[latex]3{x}^{2}[\/latex]<\/td>\r\n[latex]-x[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Find the product.\r\n

    [latex]\\left(3x+2\\right)\\left({x}^{3}-4{x}^{2}+7\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"508646\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"508646\"]\r\n\r\n[latex]3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14[\/latex]\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe can use a table to keep track of our work as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\r\n\r\n\r\n\r\n\r\n\r\n
    <\/td>\r\n[latex]3{x}^{2}[\/latex]<\/strong><\/td>\r\n[latex]-x[\/latex]<\/strong><\/td>\r\n[latex]+4[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
    [latex]2x[\/latex]<\/strong><\/td>\r\n[latex]6{x}^{3}\\\\[\/latex]<\/td>\r\n[latex]-2{x}^{2}[\/latex]<\/td>\r\n[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]+1[\/latex]<\/strong><\/td>\r\n[latex]3{x}^{2}[\/latex]<\/td>\r\n[latex]-x[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]18872[\/ohm2_question]<\/section>\r\n

    Using FOIL to Multiply Binomials<\/h3>\r\nFor quicker multiplication, especially with [pb_glossary id=\"814\"]binomials[\/pb_glossary], we can use a handy shortcut called the FOIL method.\r\n\r\n
    It is called FOIL<\/strong> because we multiply the F<\/strong>irst terms, the O<\/strong>uter terms, the I<\/strong>nner terms, and then the L<\/strong>ast terms of each binomial.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Two Visual Example of FOIL with labels[\/caption]\r\n\r\n<\/section>The FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.\r\n\r\n
    How To: Given two binomials, Multiplying Using FOIL<\/strong>\r\n
      \r\n \t
    1. Multiply the first terms of each binomial.<\/li>\r\n \t
    2. Multiply the outer terms of the binomials.<\/li>\r\n \t
    3. Multiply the inner terms of the binomials.<\/li>\r\n \t
    4. Multiply the last terms of each binomial.<\/li>\r\n \t
    5. Add the products.<\/li>\r\n \t
    6. Combine like terms and simplify.<\/li>\r\n<\/ol>\r\n<\/section>
      Use the FOIL method to find the product of the polynomials:[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]Solution\u00a0<\/strong>Find the product of the F<\/strong>irst terms:\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" First terms[\/caption]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Outer terms[\/caption]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Inner terms[\/caption]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Last terms[\/caption]\r\n\r\nFind the product of the O<\/strong>uter terms:\r\nFind the product of the I<\/strong>nner terms:Find the product of the L<\/strong>ast terms:\r\n\r\nNow combine all the terms obtained from the FOIL method:\r\n

      [latex]6x^2+6x-54x-54[\/latex]<\/p>\r\nCombine like terms ([latex]6x-54x = -48x[\/latex]) and we have found our final simplified product:\r\n

      [latex]\\left(2x-18\\right)\\left(3x + 3\\right) = 6x^2-48x-54[\/latex]<\/p>\r\n\r\n<\/section>

      Use FOIL to find the product.[latex]\\left(x+7\\right)\\left(3x - 5\\right)[\/latex][reveal-answer q=\"603351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"603351\"][latex]3{x}^{2}+16x - 35[\/latex][\/hidden-answer]<\/section>
      [ohm2_question hide_question_numbers=1]18873[\/ohm2_question]<\/section>","rendered":"

      Multiplying Polynomials<\/h2>\n

      Multiplying polynomials is a step up from adding and subtracting them, but once you get the hang of it, it’s pretty straightforward!<\/p>\n

      To multiply polynomials, we use what’s called the distributive property<\/strong>. This means we take each term from the first polynomial and multiply it by every term in the second polynomial. After that, we just combine any like terms we find.<\/p>\n

      Distributive Property: [latex]a\\cdot(b+c) = a\\cdot b+a\\cdot c[\/latex]<\/section>\n
      How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/strong><\/p>\n
        \n
      1. Multiply each term of the first polynomial by each term of the second.<\/li>\n
      2. Combine like terms.<\/li>\n
      3. Simplify.<\/li>\n<\/ol>\n<\/section>\n
        Find the product and simplify:[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]Solution\u00a0<\/strong>[latex]\\begin{align*} (2x+1)(3x^2-x+4) & = 2x(3x^2-x+4) + 1(3x^2-x+4) & \\text{Use the distributive property} \\\\ & = (6x^3-2x^2+8x) + (3x^2-x+4) & \\text{Multiply each term} \\\\ & = 6x^3 + (-2x^2+3x^2) + (8x-x) + 4 & \\text{Combine like terms} \\\\ & = 6x^3 + x^2 + 7x + 4 & \\text{Simplify to final form} \\end{align*}[\/latex]<\/p>\n