28b690d6-ddb7-4f8d-a7a4-805f74585f29.html

<h4>Activity (20 minutes)</h4>
<p>This activity teaches students to recognize perfect square trinomials by identifying the patterns in their form. With
  practice, these trinomials can easily be factored into binomials.</p>
<h4>Launch</h4>
<p>Review the patterns shown for perfect square trinomials. Emphasize that the middle term, \(2ab\), determines the sign
  of the factored binomials.</p>
<p>Factor the first question together as a class. Ask students how they might recognize that the polynomial, \(64y^2
  &minus; 80y + 25\), is a perfect square trinomial.</p>
<ul>
  <li> \(64y^2\) is a perfect square. </li>
  <li> 25 is a perfect square. </li>
  <li> If \(a = 8y\) and \(b = 5\), then \(2ab = 80y\). </li>
  <li> Since the middle term is negative, the binomials will also have a negative sign. </li>
</ul>
<p>Allow students to solve the remaining questions individually. Before closing the activity, ask students if there is
  any confusion around recognizing and factoring perfect square trinomials.</p>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
    <p class="os-raise-extrasupport-name"><!--Extra Support Name-->Engagement:  Provide structures </p>
  </div>
  <div class="os-raise-extrasupport-body"> 
    <!--Support Content--><p>Provide students with math formula support for finding the perfect square while solving practice problems and quizzes by utilizing the STAAR Algebra I Reference Materials that is linked from the Getting Started - Students Start Here - Additional Resources.  It includes these formulas:
</p>
    <p class="os-raise-text-bold">Factoring</p>
      
   <div class="os-raise-d-flex os-raise-justify-content-between">
  <p>Perfect square trinomials</p>
  <p>\(a^2+2ab+b^2=(a+b)^2\)<br>\(a^2-2ab+b^2=(a-b)^2\)
  </p>
</div>
      
         <div class="os-raise-d-flex os-raise-justify-content-between">
  <p>Difference of squares</p>
  <p>\(a^2-b^2=(a-b)(a+b)\)</p>
</div>
    <p class="os-raise-text-italicize"><!--Support Italics-->Supports accessibility for Conceptual Processing, Memory</p>

  </div>
</div>
<br>
<h4>Student Activity </h4>
<p>Remember the patterns for perfect square trinomials:</p>
<p>\(a^2 + 2ab + b^2 = (a + b)^2\)</p>
<p>\(a^2 &minus; 2ab + b^2 = (a &minus; b)^2\)</p>
<p>Factor each perfect square trinomial. If necessary, factor out the GCF first. Use the "^" symbol to enter an exponent.</p>
<ol class="os-raise-noindent">
  <li> \(64y^2 &minus; 80y + 25\) </li>
</ol>
<p><strong>Answer:</strong> \((8y &minus; 5)^2\)</p>
<ol class="os-raise-noindent" start="2">
  <li> \(16y^2 + 24y + 9\) </li>
</ol>
<p><strong>Answer:</strong> \((4y + 3)^2\)</p>
<ol class="os-raise-noindent" start="3">
  <li> \(25n^2 &minus; 120n + 144\) </li>
</ol>
<p><strong>Answer:</strong> \((5n &minus; 12)^2\)</p>
<ol class="os-raise-noindent" start="4">
  <li> \(100y^2 &minus; 20y + 1\) </li>
</ol>
<p><strong>Answer:</strong> \((10y &minus; 1)^2\)</p>
<ol class="os-raise-noindent" start="5">
  <li> \(10jk^2 + 80jk + 160j\) </li>
</ol>
<p><strong>Answer:</strong> \(10j(k + 4)^2\)</p>
<ol class="os-raise-noindent" start="6">
  <li> \(75u^4 &minus; 30u^3v + 3u^2v^2\) </li>
</ol>
<p><strong>Answer:</strong> \(3u^2(5u &minus; v)^2\)</p>
<h4>Student Facing Extension</h4>
<h5> Extending Your Thinking</h5>
<p>Factor this perfect square trinomial.</p>
<p>\(90p^4r^2 + 300p^3qr^2 + 250p^2q^2r^2\)</p>
<p><strong>Answer:</strong> \(10p^2r^2(3p + 5q)^2\)</p>
<h4>Anticipated Misconceptions</h4>
<p>If students do not recognize a polynomial from the activity as a perfect square trinomial, remind them that they must
  first factor out the GCF from the polynomial.</p>
<h4>Activity Synthesis</h4>
<p>Invite a student to explain in their own words how to identify a perfect square trinomial.</p>
<ul>
  <li> The first and last terms are perfect squares. </li>
  <li> The middle term fits the pattern \(\pm 2ab\). </li>
</ul>
<p>Invite a student to explain in their own words how to factor a perfect square trinomial.</p>
<ul>
  <li> The trinomial is factored according to one of the following patterns: </li>
  <ul>
    <li> \(a^2 + 2ab + b^2 = (a + b)^2\) </li>
    <li> \(a^2 &minus; 2ab + b^2 = (a &minus; b)^2\) </li>
  </ul>
</ul>
<h3>6.6.2: Self Check</h3>
<p class="os-raise-text-bold os-raise-text-italicize">After the activity, students will answer the following question to check their understanding of the
concepts explored in the activity.</p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Factor: \(64m^2 + 112mn + 49n^2\).</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        \(m(8 + 7n)^2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: There is no GCF to factor first. The answer is \((8m +
          7n)^2\).
      </td>
    </tr>
    <tr>
      <td>
        \((8n + 7m)^2\)
      </td>
      <td>
       Incorrect. Let&rsquo;s try again a different way: The coefficients are incorrectly reversed. The answer is
          \((8m + 7n)^2\).
      </td>
    </tr>
    <tr>
      <td>
        \((8m &minus; 7n)^2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: The sign is incorrect. It should be addition. The answer is
          \((8m + 7n)^2\).
      </td>
    </tr>
    <tr>
      <td>
        \((8m + 7n)^2\)
      </td>
      <td>
        That&rsquo;s correct! Check yourself: Find the square to check that it matches the original polynomial.
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>6.6.2: Additional Resources</h3>
<p class="os-raise-text-bold" os-raise-text-italicize>The following content is available to students who would like more support based on their experience with
      the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it.</p>
<h4>Factoring Perfect Square Trinomials</h4>
<p>Some trinomials are perfect squares. They result from multiplying a binomial by itself. We squared a binomial using
  the Binomial Squares pattern in a previous lesson.</p>
<p>\({\style{color:red}(}{\style{color:red}a}{\style{color:red}+}{\style{color:red}b}{\style{color:red})}^{\style{color:red}2}\)<br>
  \((3x+4)^2\)
</p>
<p>\(\;\;\;\;
  {\style{color:red}a}
  ^
  {\style{color:red}2}
  \;
  {\style{color:red}+}
  \;
  {\style{color:red}2}
  \;
  {\style{color:red}\cdot}
  \;
  {\style{color:red}a}
  \;
  {\style{color:red}\cdot}
  \;
  {\style{color:red}b}
  \;
  {\style{color:red}+}
  \;
  {\style{color:red}b}
  ^
  {\style{color:red}2}
  \\{(3x)}^2+2(3x\cdot4)+4^2\)</p>
<p>\(\;\;\;\;9x^2+24x+16\)</p>
<p>The trinomial \(9x^2 + 24x + 16\) is called a perfect square trinomial. It is the square of the binomial \(3x + 4\).
</p>
<p>In this chapter, you will start with a perfect square trinomial and factor it into its prime factors.</p>
<p>You could factor this trinomial using the methods described in the last section since it is of the form \(ax^2 + bx +
  c\). But if you recognize that the first and last terms are squares and the trinomial fits the perfect square
  trinomials pattern, you will save yourself a lot of work.</p>
<p>Here is the pattern&mdash;the reverse of the binomial squares pattern.</p>

<br>
<div class="os-raise-graybox">  <p class="os-raise-text-bold"> PERFECT SQUARE TRINOMIALS PATTERN </p>
  <hr>
  <p>If \(a\) and \(b\) are real numbers</p>
  <p>\(a^2 + 2ab + b^2 = (a + b)^2\)</p>
  <p>\(a^2 &minus; 2ab + b^2 = (a &minus; b)^2\)</p>
</div>
<br>
<br>
<p>To use this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading
  coefficient is a perfect square, \(a^2\). Next, check that the last term is a perfect square, \(b^2\). Then check the
  middle term: Is it the product, \(2ab\)? If everything checks, you can easily write the factors.</p>
  <p class="os-raise-text-bold">Example 1</p>
<p>Factor: \(9x^2 + 12x + 4\).</p>
<p><strong>Step 1 -</strong> Does the trinomial fit the perfect square trinomials pattern, \(a^2 + 2ab + b^2\)?</p>
<ul>
  <li> Is the first term a perfect square? Write it as a square, \(a^2\).
    <ul>
      <li>Is \(9x^2\) a perfect square?</li>
    </ul>
  </li>
  <ul>
    <li> Yes. Write it as \((3x)^2\).</li>
  </ul>
  <li> Is the last term a perfect square? Write it as a square, \(b^2\).
    <ul>
      <li>Is 4 a perfect square?</li>
    </ul>
  </li>
  <ul>
    <li> Yes. Write it as \((2)^2\).</li>
  </ul>
  <li> Check the middle term. Is it \(2ab\)?
    <ul>
      <li>Is \(12x\) twice the product of \(3x\) and \(2^2\)? Does it match?</li>
    </ul>
  </li>
  <ul>
    <li> Yes, so we have a perfect square trinomial!</li>
  </ul>
</ul>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="176" role="presentation"
    src="https://k12.openstax.org/contents/raise/resources/10d90e7be43bd48e3e66edec4e68ccc5fe2d5ec7" width="200"></p>
<p><strong>Step 2 -</strong> Write the square of the binomial.</p>
<p>  \(9x^2+12x+4\)</p>
<p>\(a^2\;+\;2\; \cdot\; a\;\cdot\; b\; +\; b^2\)<br>
  \((
  {\style{color:red}3}
  {\style{color:red}x}
  )^2+2 \cdot {\style{color:red}3}
  {\style{color:red}x} \cdot
  {\style{color:blue}2}+{\style{color:blue}2}^2\)</p>
<p>\((a+b)^2\)<br>
  \(({\style{color:red}3}
  {\style{color:red}x}+{\style{color:blue}2})^2\)</p>
<p><strong>Step 3 -</strong> Check by multiplying.</p>
<p>  \((3x+2)^2\)</p>
<p>\((3x)^2+2 \cdot 3x \cdot2+2^2\)</p>
<p>\(9x^2+12x+4\; \checkmark\)</p>
<p>The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the
  pattern \(a^2 &minus; 2ab + b^2\), which factors to \((a &minus; b)^2\).</p>
<p>The steps are summarized here. </p>
<p class="os-raise-text-bold">How to factor perfect square trinomials:</p>
<p><strong>Step 1 -</strong> Does the trinomial fit the perfect square trinomials pattern?</p>
<ul>
  <li> Is the first term a perfect square? Write it as a square. </li>
  <li>Is the last term a perfect square? Write it as a square. </li>
  <li>Check the middle term. Is it \(2ab\)? </li>
</ul>
<p><strong>Step 2 - 
  </strong>Write the square of the binomial.</p>
<p><strong>Step 3 -</strong> Check by multiplying.</p>
<p><img alt height="172" role="presentation"
    src="https://k12.openstax.org/contents/raise/resources/04a63da1463ac8cf77a6770e1df07c319abfd5ba" width="200"></p>
<p>We&rsquo;ll work one now where the middle term is negative.</p>
<p class="os-raise-text-bold">Example 2</p>
<p>Factor: \(81y^2 &minus; 72y + 16\).</p>
<p>The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The
  middle term is negative, so the binomial square would be \((a &minus; b)^2\).</p>
<p><strong>Step 1 - 
  </strong>Are the first and last terms perfect squares?</p>
<p>  \(81y^2 &minus; 72y + 16\)<br>
  \((9y)^2\;\;\;\;\;\;\;\;\;\;\;\;(4)^2\)</p>
<p>Step 2 - 
  Check the middle term.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="84" role="presentation"
    src="https://k12.openstax.org/contents/raise/resources/c562032b374a8bd555e6a8dc87376dfbf23b2011" width="134"></p>
<p>Does it match \((a &minus; b)^2\)?<br>
  \(\;\;\;
  {\style{color:red}a}
  ^
  {\style{color:red}2}\;
  {\style{color:red}-}\;
  {\style{color:red}2} \;\;\;\; {\style{color:red}a}
  \;\;\; {\style{color:red}b}\;
  {\style{color:red}+}\;
  {\style{color:red}b}
  ^
  {\style{color:red}2}
  \)<br>
  \((9y)^2-2 \cdot 9y \cdot 4+4^2\) <br>
  Yes. </p>
<p><strong>Step 3 - 
  </strong>Write as the square of a binomial.</p>
<p>  \((9y &minus; 1)^2\)</p>
<p><strong>Step 4 - 
  </strong>Check by multiplying.</p>
<p> \((9y &minus; 1)^2\)<br>
  \((9y)^2-2 \cdot 9y \cdot 4+4^2\)<br>
  \(81y^2 &minus; 72y + 16\; \checkmark\)</p>
<p>The next example will be a perfect square trinomial with two variables.</p>
<p class="os-raise-text-bold">Example 3</p>
<p>Factor: \(36x^2 + 84xy + 49y^2\).</p>
<p><strong>Step 1 -</strong> Test each term to verify the pattern.</p>
<p>  \(\;\;\;
  {\style{color:red}a}
  ^
  {\style{color:red}2}\;
  {\style{color:red}+}\;
  {\style{color:red}2}
  \;\;\;\;\;
  {\style{color:red}(}{\style{color:red}a}{\style{color:red})} \;\;\;\;\;
  {\style{color:red}(}{\style{color:red}b}{\style{color:red})}\;
  {\style{color:red}+}\;
  {\style{color:red}b}
  ^
  {\style{color:red}2}
  \)<br>
\(6x^2+2 (6x) (7y)+(7y)^2\) </p>
<p><strong>Step 2 - 
  </strong>Factor.</p>
<p>  \((6x + 7y)^2\)</p>
<p><strong>Step 3 -</strong> Check by multiplying.</p>
<p>  \((6x + 7y)^2\)<br>
  \((6x)^2+2 \cdot 6x \cdot 7y+(7y)^2\)<br>
  \(36x^2 + 84xy + 49y^2\; \checkmark\)</p>
<p>Remember, the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a
  GCF in all three terms, and it should be factored out first. And, sometimes, once the GCF has been factored, you will
  recognize a perfect square trinomial.</p>
  <p class="os-raise-text-bold">Example 4</p>
<p>Factor: \(100x^2y &minus; 80xy + 16y\).</p>
<p><strong>Step 1 - 
  </strong>Is there a GCF? Yes, \(4y\), so factor it out.</p>
<p>  \(4y(25x^2 &minus; 20x + 4)\)</p>
<p><strong>Step 2 -</strong> Is this a perfect square trinomial? Verify the pattern.</p>
<p>  \(\;\;\;\;\;\;\;\;\;
  {\style{color:red}a}
  ^
  {\style{color:red}2}\;
  {\style{color:red}-}\;
  {\style{color:red}2}
  \;\;\;\;\;
  {\style{color:red}a}
  \;\;\;\;\;

  {\style{color:red}b}\;
  {\style{color:red}+}\;
  {\style{color:red}b}
  ^
  {\style{color:red}2}
  \)<br>
  \(4y\begin{bmatrix}{(5x)}^2-2 \cdot 5x \cdot 2+ 2^2\end{bmatrix}\\\)</p>
<p><strong>Step 3 -</strong> Write as the square of a binomial.</p>
<p>  \(4y(5x &minus; 2)^2\)<br>
  Remember: Keep the factor \(4y\) in the final product.</p>
<p><strong>Step 4 - 
  </strong>Check by multiplying.</p>
<p>  \(4y(5x &minus; 2)^2\)<br>
  \(4y[(5x)^2-2 \cdot 5x \cdot 2+2^2]\)<br>
  \(4y(25x^2-20x+4)\)<br>
  \(100x^2y &minus; 80xy + 16y\; \checkmark\)</p>
<h4>Try It: Factoring Perfect Square Trinomials</h4>
<p>Factor each perfect square trinomial. If necessary, factor out the GCF first.</p>
<ol class="os-raise-noindent">
  <li> \(4x^2 + 12x + 9\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li> \(49x^2 + 84xy + 36y^2\). </li>
</ol>
<ol class="os-raise-noindent" start="3">
  <li> \(8x^2y &minus; 24xy + 18y\) </li>
</ol>
<p>Write down your answers. Then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to factor perfect square trinomials.</p>
<ol class="os-raise-noindent">
  <li> \((2x + 3)^2\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li> \((7x + 6y)^2\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
  <li> \(2y(2x &minus; 3)^2\) </li>
</ol>