# Khan/khan-exercises

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 c606589 Add partial fraction expansion exercise xymostech authored Jan 19, 2013 1 cde9e9f Re-run the exercises through the updated cleaner. jeresig authored Apr 24, 2013 2 3 0f0c630 Clean up the exercise files using the new cleaning utility. (II) cbhl authored Apr 16, 2013 4 113bddd Update the title tags to match real display names spicyj authored Apr 11, 2013 5 Partial fraction expansion ac1415e Use requirejs for module loading spicyj authored Mar 5, 2014 6 c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 7 27 28 29
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37b79cb Add partial fraction expansion exercise xymostech authored Jan 19, 2013 32
e1f2884 Fix spacing, expr, silly problems xymostech authored Jan 20, 2013 33
37b79cb Add partial fraction expansion exercise xymostech authored Jan 20, 2013 36 randRangeNonZero(-5, 5) 37 randRangeNonZero(-5, 5) 38 randRangeNonZero(-5, 5) 39 randRangeNonZero(-5, 5) 40
095347d Add some extra helpful variables xymostech authored Jan 22, 2013 41 A + B 42 -A * D - B * C 43 expr(["+", ["*", E, "x"], F]) 37b79cb Add partial fraction expansion exercise xymostech authored Jan 20, 2013 44 expr(C === -D ? c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 45 ["+", "x^2", C*D] : 46 ["+", "x^2", ["*", -C - D, "x"], C*D] 47 ) e1f2884 Fix spacing, expr, silly problems xymostech authored Jan 20, 2013 48 expr(["+", "x", -C]) 49 expr(["+", "x", -D]) c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 50
233a0b9 Rearrange answer area content within some exercises beneater authored Nov 14, 2014 51

e687e49 Change wording xymostech authored Jan 19, 2013 52 Expand c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 53 \dfrac{NUMER}{DENOM} e687e49 Change wording xymostech authored Jan 20, 2013 54 using partial fractions. 233a0b9 Rearrange answer area content within some exercises beneater authored Nov 14, 2014 55

0f0c630 Clean up the exercise files using the new cleaning utility. (II) cbhl authored Apr 16, 2013 56
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58 A 59 B 60 C 61 D c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 62
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64 B 65 A 66 D 67 C c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 68
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0f0c630 Clean up the exercise files using the new cleaning utility. (II) cbhl authored Apr 16, 2013 70
c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 71 e9807ff Clean up the exercise files using the new cleaning utility. jeresig authored Apr 16, 2013 72
1dd8e1b Don't change the width of All The Answers beneater authored Mar 6, 2013 74 a 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 75 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 77 + 78 1dd8e1b Don't change the width of All The Answers beneater authored Mar 6, 2013 80 a 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 81
1dd8e1b Don't change the width of All The Answers beneater authored Mar 6, 2013 85 x-a 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 86 1dd8e1b Don't change the width of All The Answers beneater authored Mar 6, 2013 88 x-a 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 89
62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 73 0f0c630 Clean up the exercise files using the new cleaning utility. (II) cbhl authored Apr 16, 2013 76 79 c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 82 83 62d7d45 Lint down to <80 character lines xymostech authored Jan 21, 2013 84 87 c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 90 e9807ff Clean up the exercise files using the new cleaning utility. jeresig authored Apr 16, 2013 91 c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 92 93 94 95
b2e5967 Write real hints xymostech authored Jan 22, 2013 96

97 First, factor the denominator to find the denominators of 98 the two fractions we will split our fraction into. 99

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101 102 DENOM = 103 (ADENOM)(BDENOM) 104 105

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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

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113 df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 114 \dfrac{NUMER}{ 115 (ADENOM)(BDENOM) 116 } = b2e5967 Write real hints xymostech authored Jan 22, 2013 117 \dfrac{?}{ADENOM} + 118 \dfrac{?}{BDENOM} 119 120

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b2e5967 Write real hints xymostech authored Jan 22, 2013 122

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

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1772880 Typo denomiators -> denominators petercollingridge authored Mar 19, 2013 128 In our case, both of the denominators have a degree of b2e5967 Write real hints xymostech authored Jan 22, 2013 129 1, so we replace our numerators with 130 polynomials of degree 0, or constants. We 131 will use the constants A and 132 B. 133

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136 df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 137 \dfrac{NUMER}{ 138 (ADENOM)(BDENOM) 139 } = b2e5967 Write real hints xymostech authored Jan 22, 2013 140 \dfrac{A}{ADENOM} + 141 \dfrac{B}{BDENOM} 142 143

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145 Now, to get rid of the fractions, we multiply by the common 146 denominator, 147 (ADENOM)(BDENOM). 148

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c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 150 b2e5967 Write real hints xymostech authored Jan 22, 2013 151 NUMER = 152 A(BDENOM) + B(ADENOM) c606589 Add partial fraction expansion exercise xymostech authored Jan 20, 2013 153 b2e5967 Write real hints xymostech authored Jan 22, 2013 154

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156 Now we can solve for A and B. An 157 easy way to do this is to try to choose values for 158 x that will get one of A or 159 B to cancel out, and then solve for the other 160 one. 161

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164 Let's try to cancel out B. We see that if 165 we plug in C for 166 x, the term with B cancels 167 out, and we are left with: 168

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df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 170 171 expr(["+", E * C, F]) = 172 A(expr(["+", C, -D])) 173 b2e5967 Write real hints xymostech authored Jan 22, 2013 174

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df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 177 178 E * C + F = 179 expr(["*", C - D, "A"]) 180 b2e5967 Write real hints xymostech authored Jan 22, 2013 181

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183 A=A 184

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187 We can do the same thing to solve for B, 188 but instead plugging in D for 189 x: 190

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df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 192 193 expr(["+", E * D, F]) = 194 B(expr(["+", D, -C])) 195 b2e5967 Write real hints xymostech authored Jan 22, 2013 196

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df4d307 Lint down to <80 character lines xymostech authored Jan 22, 2013 199 200 E * D + F = 201 expr(["*", D - C, "B"]) 202 b2e5967 Write real hints xymostech authored Jan 22, 2013 203

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205 B=B 206

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209 Now, we plug back in to our fractions, and get: 210

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212 213 \dfrac{NUMER}{DENOM} = 214 \dfrac{A}{ADENOM} + 215 \dfrac{B}{BDENOM} 216 217

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220 221 222 cde9e9f Re-run the exercises through the updated cleaner. jeresig authored Apr 24, 2013 223 224
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