3.10:_Division_Explorations

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<p class="lt-math-10384">Anu refuses to tell anyone if she is working in a 1&larr;10 system, or a 1&larr;5 system, or any other system.  She makes everyone call it a 1 &larr; <i>x</i> system but won&rsquo;t tell anyone what <i>x</i> stands for.</p>

<p class="lt-math-10384">We know that boxes in a 1&larr;10 have values that are powers of ten: 1, 10, 100, 1000, 10000&hellip;</p>

<p class="lt-math-10384">And boxes in a 1&larr;5 system are powers of five: 1, 5, 25, 125, 625&hellip;</p>

<p class="lt-math-10384">So Anu&rsquo;s system, whatever it is, must be powers of <em>x</em>: $1, x, x^{2}, x^{3}, x^{4} \ldots$</p>

<p class="lt-math-10384">\[2x^{3} + 5x^{2} + 5x + 6 \ldotp \nonumber \]</p>

<p class="lt-math-10384">And when she writes $12_{x}$ she means:</p>

<p class="lt-math-10384">\[x +  2 \ldotp \nonumber \]</p>

<p class="lt-math-10384">Anu decides to compute $2556_{x} \div 12_{x}$.</p>

<p class="lt-math-10384">She obtains:</p>

<p class="lt-math-10384">\[(2x^{3} + 5x^{2} + 5x + 6) \div (x + 2) = 2x^{2} + x + 3 \ldotp \nonumber \]</p>

<section class="box-example"><span id="Problem_28"></span><h2 class="box-legend lt-math-10384"><span class="lt-icon-default">Problem 28</span></h2>

<ol>
    <li class="lt-math-10384">Check Anu&rsquo;s division by computing the product $$(x + 2)(2x^{2} + x + 3) \ldotp$$Did it work?</li>
    <li class="lt-math-10384">Use Anu&rsquo;s method to find $$(3x^{2} + 7x + 2) \div (x + 2) \ldotp$$</li>
    <li class="lt-math-10384">Use Anu&rsquo;s method to find $$(2x^{4} + 3x^{3} + 5x^{2} + 4x + 1) \div (2x + 1) \ldotp$$</li>
    <li class="lt-math-10384">Use Anu&rsquo;s method to find $$(x^{4} + 3x^{3} + 6x^{2} + 5x + 3) \div (x^{2} + x + 1) \ldotp$$</li>
</ol>
</section>

<p class="lt-math-10384">Anu later tells use that she really was thinking of a 1&larr;10 system so that <i>x</i>does equal ten. Then her number 2556 really was two thousand, five hundred and fifty six and 12 really was twelve. Her statement:</p>

<p class="lt-math-10384">\[(2x^{3} + 5x^{2} + 5x + 6) \div (x + 2) = 2x^{2} + x + 3 \ldotp \nonumber \]</p>

<p class="lt-math-10384">is actually 2556 : 12 = 213.</p>

<section class="box-example"><span id="Problem_29"></span><h2 class="box-legend lt-math-10384"><span class="lt-icon-default">Problem 29</span></h2>

<ol>
    <li class="lt-math-10384">Check that 2556 : 12 = 213 is correct in base 10.</li>
    <li class="lt-math-10384">Keeping with the 1&larr;10 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that your answers are correct.</li>
</ol>
</section>

<p class="lt-math-10384">Uh Oh! Anu has changed her mind. She now says she was thinking of a 1&larr;11 system.</p>

<p class="lt-math-10384">Now $2556_{x}$ means $2 \cdot 11^{3} + 5 \cdot 11^{2} + 5 \cdot 11 + 6 = 3328_{ten}$.</p>

<p class="lt-math-10384">Similarly, $12_{x}$ means $1 \cdot 11 + 2 = 13_{ten}$, and $213_{x}$ means $2 \cdot 11^{2} + 1 \cdot 11 + 3 = 256_{ten}$.</p>

<p class="lt-math-10384">So Anu&rsquo;s computation $2556_{x} \div 12_{x}\ = 213_{x}$ is actually the (base 10) statement: \[3328 : 13 = 256 \ldotp \nonumber \]</p>

<section class="box-example"><span id="Problem_30"></span><h2 class="box-legend lt-math-10384"><span class="lt-icon-default">Problem 30</span></h2>

<ol>
    <li class="lt-math-10384">Check that 3328 : 13 = 256 is also correct in base ten.</li>
    <li class="lt-math-10384">Keeping with the 1&larr;11 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that they are correct.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_31"></span><h2 class="box-legend lt-math-10384"><span class="lt-icon-default">Problem 31</span></h2>

<ol>
    <li class="lt-math-10384">Use Anu&rsquo;s method to show that $$(x^{4} + 4x^{3} + 6x^{2} + 4x + 1) \div (x + 1) = (x^{3} + 3x^{2} + 3x + 1) \ldotp$$</li>
    <li class="lt-math-10384">What is this saying for <i>x</i> = 10?  Check that the division is correct.</li>
    <li class="lt-math-10384">What is this saying for <i>x</i> = 2? Check that the division is correct.</li>
    <li class="lt-math-10384">What is this saying for <i>x</i> equal to each of 3, 4, 5, 6, 7, 8, 9, and 11? Check that each division is correct.</li>
    <li class="lt-math-10384">What is this saying for <em>x</em> = 0?</li>
</ol>
</section>

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