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<!DOCTYPE html>
<html data-require="math math-format expressions rational-expressions">
<head>
    <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
    <title>Simplifying rational expressions 3</title>
    <script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
    <div class="exercise">
        <div class="problems">
            <div>
                <div class="vars">
                    <var id="X">randVar()</var>
                    <var id="Y" data-ensure="X !== Y">randVar()</var>
                    <var id="A">randRangeNonZero(-10, 10)</var>
                    <div data-ensure="A !== C && B !== C && (B !== 0 || C !== 0)">
                        <var id="B">randRangeWeighted(-10, 10, 0, 0.5)</var>
                        <var id="C">randRangeWeighted(-10, 10, 0, 0.5)</var>
                    </div>
                    <var id="FACTOR1">new RationalExpression([[1, X], A])</var>
                    <var id="FACTOR2">new RationalExpression([[1, X], B])</var>
                    <var id="FACTOR3">new RationalExpression([[1, X], C])</var>
                    <var id="NUMERATOR">FACTOR1.multiply(FACTOR2)</var>
                    <var id="DENOMINATOR">FACTOR1.multiply(FACTOR3)</var>
                </div>

                <p class="problem">Simplify the following expression and state the condition under which the simplification is valid:</p>
                <p class="question"><code><var>Y</var> = \dfrac{<var>NUMERATOR</var>}{<var>DENOMINATOR</var>}</code></p>

                <div class="solution" data-type="set">
                    <div class="set-sol" data-type="multiple">
                        <span class="sol" data-type="regex"><var>FACTOR2.regex()</var></span>
                        <span class="sol" data-type="regex"><var>FACTOR3.regex()</var></span>
                        <span class="sol" data-type="number"><var>-A</var></span>
                    </div>
                    <div class="set-sol" data-type="multiple">
                        <span class="sol" data-type="regex"><var>FACTOR2.multiply(-1).regex()</var></span>
                        <span class="sol" data-type="regex"><var>FACTOR3.multiply(-1).regex()</var></span>
                        <span class="sol" data-type="number"><var>-A</var></span>
                    </div>

                    <div class="input-format">
                        <div class="entry" data-type="multiple">
                            <table class="rational-exp">
                                <tbody><tr>
                                    <td>
                                        <code><var>Y</var> = </code>
                                    </td>
                                    <td><table>
                                        <tbody><tr>
                                            <td class="soln-top">
                                                <span class="sol short40">a</span>
                                            </td>
                                        </tr>
                                        <tr>
                                            <td class="soln-bot">
                                                <span class="sol short40" data-fallback="1">a</span>
                                            </td>
                                        </tr>
                                    </tbody></table></td>
                                    <td>
                                        <code> \space <var>X</var> \neq </code><span class="sol soln-dom short32">a</span>
                                    </td>
                                </tr>
                            </tbody></table>
                        </div>
                    </div>
                    <div class="example">a simplifed expression, like <code>x + 2</code></div>
                </div>

                <div class="hints">
                    <p>First factor the expressions in the numerator and denominator.</p>

                    <p><code>
                        \dfrac{<var>NUMERATOR</var>}{<var>DENOMINATOR</var>}
                        = \dfrac{(<var>FACTOR2</var>)(<var>FACTOR1</var>)}{(<var>FACTOR3</var>)(<var>FACTOR1</var>)}
                    </code></p>

                    <p>Notice that the term <code>(<var>FACTOR1</var>)</code> appears in both the numerator and denominator.</p>

                    <div>
                        <p>Dividing both the numerator and denominator by <code>(<var>FACTOR1</var>)</code> gives:</p>
                        <div data-if="FACTOR2.toString() === '1'">
                            <code><var>Y</var> = \dfrac{<var>FACTOR2</var>}{<var>FACTOR3</var>}</code> or more simply,
                            <code><var>Y</var> = <var>FACTOR2</var></code>
                        </div>
                        <div data-else="">
                            <code><var>Y</var> = \dfrac{<var>FACTOR2</var>}{<var>FACTOR3</var>}</code>
                        </div>
                    </div>

                    <div>
                        <p>Since we divided by <code>(<var>FACTOR1</var>)</code>, <code><var>X</var> \neq <var>-A</var></code>.</p>
                        <p><code><var>Y</var> = <var>writeExpressionFraction(FACTOR2, FACTOR3)</var>; \space <var>X</var> \neq <var>-A</var></code></p>
                    </div>
                </div>
            </div>
        </div>
    </div>
</body>
</html>
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