{"id":868,"date":"2024-04-30T00:50:26","date_gmt":"2024-04-30T00:50:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=868"},"modified":"2024-11-20T02:43:37","modified_gmt":"2024-11-20T02:43:37","slug":"factoring-polynomials-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/factoring-polynomials-learn-it-3\/","title":{"raw":"Factoring Polynomials: Learn It 3","rendered":"Factoring Polynomials: Learn It 3"},"content":{"raw":"

Factoring a Perfect Square Trinomial<\/h2>\r\nA perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.\r\n
[latex]\\begin{array}{ccc}\\hfill {a}^{2}+2ab+{b}^{2}& =& {\\left(a+b\\right)}^{2}\\hfill \\\\ & \\text{and}& \\\\ \\hfill {a}^{2}-2ab+{b}^{2}& =& {\\left(a-b\\right)}^{2}\\hfill \\end{array}[\/latex]<\/div>\r\n
We can use this equation to factor any perfect square trinomial.<\/div>\r\n
\r\n

perfect square trinomial<\/h3>\r\nA perfect square trinomial can be written as the square of a binomial:\r\n
[latex]{a}^{2}+2ab+{b}^{2}={\\left(a+b\\right)}^{2}[\/latex]<\/div>\r\n<\/section>
How To: Given a perfect square trinomial, factor it into the square of a binomial\r\n<\/strong>\r\n
    \r\n \t
  1. Confirm that the first and last term are perfect squares.<\/li>\r\n \t
  2. Confirm that the middle term is twice the product of [latex]ab[\/latex].<\/li>\r\n \t
  3. Write the factored form as [latex]{\\left(a+b\\right)}^{2}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Factor [latex]25{x}^{2}+20x+4[\/latex].Solution<\/strong>\r\n\r\nTo factor the quadratic expression [latex]25{x}^{2}+20x+4[\/latex], recognizing it as a perfect square trinomial will streamline the process. This type of expression comes from squaring a binomial and has a special format, [latex]a^2 +2ab+b^2[\/latex], where it can be rewritten as [latex](a+b)^2[\/latex].[latex]\\begin{align*} \\text{Original expression:} & \\quad 25x^2 + 20x + 4 \\\\ \\text{Identify square terms:} & \\quad 25x^2 = (5x)^2 \\quad \\text{and} \\quad 4 = 2^2 \\\\ \\text{Check middle term:} & \\quad 2 \\times 5x \\times 2 = 20x \\\\ \\text{Write as a square of a binomial:} & \\quad (5x + 2)^2 \\end{align*}[\/latex]\r\n\r\n<\/section>
    Factor [latex]49{x}^{2}-14x+1[\/latex].[reveal-answer q=\"530221\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"530221\"][latex]{\\left(7x - 1\\right)}^{2}[\/latex][\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]18884[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]18885[\/ohm2_question]<\/section>\r\n

    Factoring a Difference of Squares<\/h2>\r\nA difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.\r\n
    [latex]{a}^{2}-{b}^{2}=\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/div>\r\nWe can use this equation to factor any differences of squares.\r\n\r\n
    \r\n

    difference of squares<\/h3>\r\nA difference of squares can be rewritten as two factors containing the same terms but opposite signs.\r\n
    [latex]{a}^{2}-{b}^{2}=\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/div>\r\n<\/section>
    How To: Given a difference of squares, factor it into binomials<\/strong>\r\n
      \r\n \t
    1. Confirm that the first and last term are perfect squares.<\/li>\r\n \t
    2. Write the factored form as [latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
      Factor [latex]9{x}^{2}-25[\/latex].Solution<\/strong>\r\n\r\nTo factor the quadratic expression [latex]9{x}^{2}-25[\/latex], we recognize that it is a difference of squares.[latex]\\begin{align*} \\text{Original expression:} & \\quad 9x^2 - 25 \\\\ \\text{Identify square terms:} & \\quad 9x^2 = (3x)^2 \\quad \\text{and} \\quad 25 = 5^2 \\\\ \\text{Apply difference of squares formula:} & \\quad (3x + 5)(3x - 5) \\end{align*}[\/latex]\r\n\r\n<\/section>
      Factor [latex]81{y}^{2}-100[\/latex].[reveal-answer q=\"979538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979538\"][latex]\\left(9y+10\\right)\\left(9y - 10\\right)[\/latex][\/hidden-answer]<\/section>
      [ohm2_question hide_question_numbers=1]18886[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]18887[\/ohm2_question]<\/section>","rendered":"

      Factoring a Perfect Square Trinomial<\/h2>\n

      A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.<\/p>\n

      [latex]\\begin{array}{ccc}\\hfill {a}^{2}+2ab+{b}^{2}& =& {\\left(a+b\\right)}^{2}\\hfill \\\\ & \\text{and}& \\\\ \\hfill {a}^{2}-2ab+{b}^{2}& =& {\\left(a-b\\right)}^{2}\\hfill \\end{array}[\/latex]<\/div>\n
      We can use this equation to factor any perfect square trinomial.<\/div>\n
      \n

      perfect square trinomial<\/h3>\n

      A perfect square trinomial can be written as the square of a binomial:<\/p>\n

      [latex]{a}^{2}+2ab+{b}^{2}={\\left(a+b\\right)}^{2}[\/latex]<\/div>\n<\/section>\n
      How To: Given a perfect square trinomial, factor it into the square of a binomial
      \n<\/strong><\/p>\n
        \n
      1. Confirm that the first and last term are perfect squares.<\/li>\n
      2. Confirm that the middle term is twice the product of [latex]ab[\/latex].<\/li>\n
      3. Write the factored form as [latex]{\\left(a+b\\right)}^{2}[\/latex].<\/li>\n<\/ol>\n<\/section>\n
        Factor [latex]25{x}^{2}+20x+4[\/latex].Solution<\/strong><\/p>\n

        To factor the quadratic expression [latex]25{x}^{2}+20x+4[\/latex], recognizing it as a perfect square trinomial will streamline the process. This type of expression comes from squaring a binomial and has a special format, [latex]a^2 +2ab+b^2[\/latex], where it can be rewritten as [latex](a+b)^2[\/latex].[latex]\\begin{align*} \\text{Original expression:} & \\quad 25x^2 + 20x + 4 \\\\ \\text{Identify square terms:} & \\quad 25x^2 = (5x)^2 \\quad \\text{and} \\quad 4 = 2^2 \\\\ \\text{Check middle term:} & \\quad 2 \\times 5x \\times 2 = 20x \\\\ \\text{Write as a square of a binomial:} & \\quad (5x + 2)^2 \\end{align*}[\/latex]<\/p>\n<\/section>\n

        Factor [latex]49{x}^{2}-14x+1[\/latex].<\/p>\n