Skip to content
Newer
Older
100644 225 lines (224 sloc) 9.04 KB
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
1 <!DOCTYPE html>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
jeresig authored
2 <html data-require="math expressions">
3 <head>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
cbhl authored
4 <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
113bddd @spicyj Update the title tags to match real display names
spicyj authored
5 <title>Partial fraction expansion</title>
ac1415e @spicyj Use requirejs for module loading
spicyj authored
6 <script data-main="../local-only/main.js" src="../local-only/require.js"></script>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
7 <style type="text/css">
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
8 #solutionarea .soln-bot input[type="text"] {
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
9 margin-left: 3px;
10 }
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
11 #solutionarea td {
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
12 text-align: center;
13 vertical-align: middle;
14 padding: 0px;
15 }
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
16 #solutionarea .soln-top {
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
17 padding-bottom: 1px;
18 }
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
19 #solutionarea .soln-middle {
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
20 padding: 0px 5px;
21 }
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
22 #solutionarea .soln-bot {
e1f2884 @xymostech Fix spacing, expr, silly problems
xymostech authored
23 padding-top: 1px;
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
24 border-top: 1px solid black;
25 }
26 </style>
27 </head>
28 <body>
29 <div class="exercise">
30 <div class="problems">
31 <div>
37b79cb @xymostech Add partial fraction expansion exercise
xymostech authored
32 <div class="vars">
e1f2884 @xymostech Fix spacing, expr, silly problems
xymostech authored
33 <div data-ensure="A + B !== 0 &amp;&amp;
34 A*D + B*C !== 0 &amp;&amp;
35 C !== D">
37b79cb @xymostech Add partial fraction expansion exercise
xymostech authored
36 <var id="A">randRangeNonZero(-5, 5)</var>
37 <var id="B">randRangeNonZero(-5, 5)</var>
38 <var id="C">randRangeNonZero(-5, 5)</var>
39 <var id="D">randRangeNonZero(-5, 5)</var>
40 </div>
095347d @xymostech Add some extra helpful variables
xymostech authored
41 <var id="E">A + B</var>
42 <var id="F">-A * D - B * C</var>
43 <var id="NUMER">expr(["+", ["*", E, "x"], F])</var>
37b79cb @xymostech Add partial fraction expansion exercise
xymostech authored
44 <var id="DENOM">expr(C === -D ?
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
45 ["+", "x^2", C*D] :
46 ["+", "x^2", ["*", -C - D, "x"], C*D]
47 )</var>
e1f2884 @xymostech Fix spacing, expr, silly problems
xymostech authored
48 <var id="ADENOM">expr(["+", "x", -C])</var>
49 <var id="BDENOM">expr(["+", "x", -D])</var>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
50 </div>
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
51 <p class="question">
e687e49 @xymostech Change wording
xymostech authored
52 Expand
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
53 <code>\dfrac{<var>NUMER</var>}{<var>DENOM</var>}</code>
e687e49 @xymostech Change wording
xymostech authored
54 using partial fractions.
233a0b9 @beneater Rearrange answer area content within some exercises
beneater authored
55 </p>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
cbhl authored
56 <div class="solution" data-type="set">
57 <div class="set-sol" data-type="multiple">
58 <span class="sol" data-forms="integer"><var>A</var></span>
59 <span class="sol" data-forms="integer"><var>B</var></span>
60 <span class="sol" data-forms="integer"><var>C</var></span>
61 <span class="sol" data-forms="integer"><var>D</var></span>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
62 </div>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
cbhl authored
63 <div class="set-sol" data-type="multiple">
64 <span class="sol" data-forms="integer"><var>B</var></span>
65 <span class="sol" data-forms="integer"><var>A</var></span>
66 <span class="sol" data-forms="integer"><var>D</var></span>
67 <span class="sol" data-forms="integer"><var>C</var></span>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
68 </div>
69 <div class="input-format">
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
cbhl authored
70 <div class="entry" data-type="multiple">
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
71 <table>
e9807ff @jeresig Clean up the exercise files using the new cleaning utility.
jeresig authored
72 <tbody><tr>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
73 <td class="soln-top">
1dd8e1b @beneater Don't change the width of All The Answers
beneater authored
74 <span class="sol short35">a</span>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
75 </td>
0f0c630 @cbhl Clean up the exercise files using the new cleaning utility. (II)
cbhl authored
76 <td class="soln-middle" rowspan="2">
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
77 <code>+</code>
78 </td>
79 <td class="soln-top">
1dd8e1b @beneater Don't change the width of All The Answers
beneater authored
80 <span class="sol short35">a</span>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
81 </td>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
82 </tr>
83 <tr>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
84 <td class="soln-bot">
1dd8e1b @beneater Don't change the width of All The Answers
beneater authored
85 <code>x-</code><span class="sol short35">a</span>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
86 </td>
87 <td class="soln-bot">
1dd8e1b @beneater Don't change the width of All The Answers
beneater authored
88 <code>x-</code><span class="sol short35">a</span>
62d7d45 @xymostech Lint down to <80 character lines
xymostech authored
89 </td>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
90 </tr>
e9807ff @jeresig Clean up the exercise files using the new cleaning utility.
jeresig authored
91 </tbody></table>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
92 </div>
93 </div>
94 </div>
95 <div class="hints">
b2e5967 @xymostech Write real hints
xymostech authored
96 <p>
97 First, factor the denominator to find the denominators of
98 the two fractions we will split our fraction into.
99 </p>
100 <p>
101 <code>
102 <var>DENOM</var> =
103 (<var>ADENOM</var>)(<var>BDENOM</var>)
104 </code>
105 </p>
106 <p>
107 Because the original denominator can be factored into these
108 two parts, we can write out our original fraction as the
109 sum of two fractions whose denominators are the two factors
110 we just found.
111 </p>
112 <p>
113 <code>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
114 \dfrac{<var>NUMER</var>}{
115 (<var>ADENOM</var>)(<var>BDENOM</var>)
116 } =
b2e5967 @xymostech Write real hints
xymostech authored
117 \dfrac{?}{<var>ADENOM</var>} +
118 \dfrac{?}{<var>BDENOM</var>}
119 </code>
120 </p>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
121 <div>
b2e5967 @xymostech Write real hints
xymostech authored
122 <p>
123 Now, we replace the numerators with polynomials of a
124 degree one less than the degree of the polynomial in
125 the denominator.
126 </p>
127 <p>
1772880 @petercollingridge Typo denomiators -> denominators
petercollingridge authored
128 In our case, both of the denominators have a degree of
b2e5967 @xymostech Write real hints
xymostech authored
129 <code>1</code>, so we replace our numerators with
130 polynomials of degree <code>0</code>, or constants. We
131 will use the constants <code>A</code> and
132 <code>B</code>.
133 </p>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
134 </div>
b2e5967 @xymostech Write real hints
xymostech authored
135 <p>
136 <code>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
137 \dfrac{<var>NUMER</var>}{
138 (<var>ADENOM</var>)(<var>BDENOM</var>)
139 } =
b2e5967 @xymostech Write real hints
xymostech authored
140 \dfrac{A}{<var>ADENOM</var>} +
141 \dfrac{B}{<var>BDENOM</var>}
142 </code>
143 </p>
144 <p>
145 Now, to get rid of the fractions, we multiply by the common
146 denominator,
147 <code>(<var>ADENOM</var>)(<var>BDENOM</var>)</code>.
148 </p>
149 <p>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
150 <code>
b2e5967 @xymostech Write real hints
xymostech authored
151 <var>NUMER</var> =
152 A(<var>BDENOM</var>) + B(<var>ADENOM</var>)
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
153 </code>
b2e5967 @xymostech Write real hints
xymostech authored
154 </p>
155 <p>
156 Now we can solve for <code>A</code> and <code>B</code>. An
157 easy way to do this is to try to choose values for
158 <code>x</code> that will get one of <code>A</code> or
159 <code>B</code> to cancel out, and then solve for the other
160 one.
161 </p>
162 <div>
163 <p>
164 Let's try to cancel out <code>B</code>. We see that if
165 we plug in <code><var>C</var></code> for
166 <code>x</code>, the term with <code>B</code> cancels
167 out, and we are left with:
168 </p>
169 <p>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
170 <code>
171 <var>expr(["+", E * C, F])</var> =
172 A(<var>expr(["+", C, -D])</var>)
173 </code>
b2e5967 @xymostech Write real hints
xymostech authored
174 </p>
175 </div>
176 <p>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
177 <code>
178 <var>E * C + F</var> =
179 <var>expr(["*", C - D, "A"])</var>
180 </code>
b2e5967 @xymostech Write real hints
xymostech authored
181 </p>
182 <p>
183 <code>A=<var>A</var></code>
184 </p>
185 <div>
186 <p>
187 We can do the same thing to solve for <code>B</code>,
188 but instead plugging in <code><var>D</var></code> for
189 <code>x</code>:
190 </p>
191 <p>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
192 <code>
193 <var>expr(["+", E * D, F])</var> =
194 B(<var>expr(["+", D, -C])</var>)
195 </code>
b2e5967 @xymostech Write real hints
xymostech authored
196 </p>
197 </div>
198 <p>
df4d307 @xymostech Lint down to <80 character lines
xymostech authored
199 <code>
200 <var>E * D + F</var> =
201 <var>expr(["*", D - C, "B"])</var>
202 </code>
b2e5967 @xymostech Write real hints
xymostech authored
203 </p>
204 <p>
205 <code>B=<var>B</var></code>
206 </p>
207 <div>
208 <p>
209 Now, we plug back in to our fractions, and get:
210 </p>
211 <p>
212 <code>
213 \dfrac{<var>NUMER</var>}{<var>DENOM</var>} =
214 \dfrac{<var>A</var>}{<var>ADENOM</var>} +
215 \dfrac{<var>B</var>}{<var>BDENOM</var>}
216 </code>
217 </p>
c606589 @xymostech Add partial fraction expansion exercise
xymostech authored
218 </div>
219 </div>
220 </div>
221 </div>
222 </div>
cde9e9f @jeresig Re-run the exercises through the updated cleaner.
jeresig authored
223 </body>
224 </html>
Something went wrong with that request. Please try again.