1 | test_roots = ->
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2 | run_test [
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3 |
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4 | "roots(x)",
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5 | "0",
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6 |
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7 | "roots(2^x-y,y)",
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8 | "2^x",
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9 |
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10 | "roots(x^2)",
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11 | "0",
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12 |
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13 | "roots(x^3)",
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14 | "0",
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15 |
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16 | "roots(2 x)",
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17 | "0",
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18 |
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19 | "roots(2 x^2)",
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20 | "0",
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21 |
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22 | "roots(i*x^2-13*i*x+36*i)",
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23 | "[4,9]",
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24 |
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25 | "roots(2 x^3)",
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26 | "0",
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27 |
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28 | "roots(6+11*x+6*x^2+x^3)",
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29 | "[-3,-2,-1]",
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30 |
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31 | "roots(a*x^2+b*x+c)",
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32 |
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33 | "[-1/2*(b^2/(a^2)-4*c/a)^(1/2)-b/(2*a),1/2*(b^2/(a^2)-4*c/a)^(1/2)-b/(2*a)]",
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34 |
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35 | "roots(3+7*x+5*x^2+x^3)",
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36 | "[-3,-1]",
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37 |
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38 | "roots(x^3+x^2+x+1)",
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39 | "[-1,-i,i]",
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40 |
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41 | "roots(x^2==1)",
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42 | "[-1,1]",
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43 |
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44 | "roots(3 x + 12 == 24)",
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45 | "4",
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46 |
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47 | "y=roots(x^2+b*x+c/k)[1]",
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48 | "",
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49 |
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50 | "y^2+b*y+c/k",
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51 | "0",
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52 |
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53 | "y=roots(x^2+b*x+c/k)[2]",
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54 | "",
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55 |
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56 | "y^2+b*y+c/k",
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57 | "0",
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58 |
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59 | "y=roots(a*x^2+b*x+c/4)[1]",
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60 | "",
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61 |
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62 | "a*y^2+b*y+c/4",
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63 | "0",
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64 |
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65 | "y=roots(a*x^2+b*x+c/4)[2]",
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66 | "",
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67 |
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68 | "a*y^2+b*y+c/4",
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69 | "0",
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70 |
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71 | "y=quote(y)",
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72 | "",
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73 |
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74 |
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75 |
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76 |
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77 |
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78 |
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79 |
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80 |
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81 |
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82 | "roots(x^3 + x^2 + x + 1)",
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83 | "[-1,-i,i]",
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84 |
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85 | "roots(2*x^3 + 3*x^2 - 11*x - 6)",
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86 | "[-3,-1/2,2]",
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87 |
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88 | "roots(x^3 - 7*x^2 + 4*x + 12)",
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89 | "[-1,2,6]",
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90 |
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91 | "roots(x^3 + 1)",
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92 | "[-1,1/2-1/2*i*3^(1/2),1/2+1/2*i*3^(1/2)]",
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93 |
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94 |
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95 | "roots(x^3 - 1)",
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96 | "[1,-1/2-1/2*i*3^(1/2),-1/2+1/2*i*3^(1/2)]",
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97 |
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98 |
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99 | "clearall",
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100 | "",
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101 |
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102 |
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103 | "thePoly = x^3 + d",
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104 | "",
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105 |
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106 | "roots(thePoly)",
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107 |
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108 |
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109 |
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110 | "[1/2*d^(1/3)-1/2*i*3^(1/2)*d^(1/3),1/2*d^(1/3)+1/2*i*3^(1/2)*d^(1/3),-d^(1/3)]",
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111 |
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112 | "and((simplify(subst(last[1],x,thePoly)) == 0),(simplify(subst(last[2],x,thePoly)) == 0),(simplify(subst(last[3],x,thePoly)) == 0))",
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113 | "1",
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114 |
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115 |
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116 |
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117 |
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118 |
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119 |
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120 |
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121 | "clearall",
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122 | "",
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123 |
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124 | "thePoly = a*x^3 + b*x^2 + c*x + d",
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125 | "",
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126 |
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127 | "roots(thePoly)",
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128 | "[-1/3*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-b^2/(3*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3))-b/(3*a)+c/(a*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)),(-1/3*a*b*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-1/2*a*c+1/6*b^2-1/2*i*3^(1/2)*a*c-1/6*i*3^(1/2)*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3)+1/6*i*3^(1/2)*b^2+1/6*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3))/(a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)),(-1/3*a*b*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3)-1/2*a*c+1/6*b^2+1/2*i*3^(1/2)*a*c+1/6*i*3^(1/2)*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3)-1/6*i*3^(1/2)*b^2+1/6*a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(2/3))/(a^2*(1/2*(-27*b^2*c^2/(a^4)+108*b^3*d/(a^4)-486*b*c*d/(a^3)+108*c^3/(a^3)+729*d^2/(a^2))^(1/2)+b^3/(a^3)-9*b*c/(2*a^2)+27*d/(2*a))^(1/3))]",
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129 |
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130 | "and((simplify(subst(last[1],x,thePoly)) == 0),(simplify(subst(last[2],x,thePoly)) == 0),(simplify(subst(last[3],x,thePoly)) == 0))",
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131 | "1",
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132 |
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133 | "roots(x^3 + 6x - 20)",
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134 | "[2,-1-3*i,-1+3*i]",
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135 |
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136 | "roots(x^3 - 6x - 40)",
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137 | "[4,-2-i*2^(1/2)*3^(1/2),-2+i*2^(1/2)*3^(1/2)]",
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138 |
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139 | "roots(x^3 + x^2 - 5x - 5)",
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140 | "[-1,-5^(1/2),5^(1/2)]",
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141 |
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142 | "roots(x^3 - 8x + 3)",
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143 | "[-3,3/2-1/2*5^(1/2),3/2+1/2*5^(1/2)]",
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144 |
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145 | "roots(x^3 - 8x - 3)",
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146 | "[3,-3/2-1/2*5^(1/2),-3/2+1/2*5^(1/2)]",
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147 |
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148 | "roots(x^3 - 18x + 35)",
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149 | "[-5,5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]",
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150 |
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151 | "clearall",
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152 | "",
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153 |
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154 |
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155 | "thePoly = x^3 - 3x + 1",
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156 | "",
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157 |
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158 | "roots(thePoly)",
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159 | "[-(-1)^(1/9)+(-1)^(8/9),1/2*cos(1/9*pi)-1/2*cos(8/9*pi)+1/2*i*sin(1/9*pi)-1/2*i*sin(8/9*pi)-3^(1/2)*cos(11/18*pi),1/2*cos(1/9*pi)-1/2*cos(8/9*pi)+1/2*i*sin(1/9*pi)-1/2*i*sin(8/9*pi)+3^(1/2)*cos(11/18*pi)]",
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160 |
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161 |
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162 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))",
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163 | "1",
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164 |
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165 | "clearall",
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166 | "",
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167 |
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168 |
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169 | "thePoly = x^3 - 3x - 1",
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170 | "",
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171 |
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172 | "roots(thePoly)",
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173 | "[-(-1)^(2/9)+(-1)^(7/9),1/2*cos(2/9*pi)-1/2*cos(7/9*pi)+1/2*i*sin(2/9*pi)-1/2*i*sin(7/9*pi)-3^(1/2)*cos(13/18*pi),1/2*cos(2/9*pi)-1/2*cos(7/9*pi)+1/2*i*sin(2/9*pi)-1/2*i*sin(7/9*pi)+3^(1/2)*cos(13/18*pi)]",
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174 |
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175 |
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176 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))",
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177 | "1",
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178 |
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179 | "roots(x^3 - 15x - 4)",
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180 | "[4,-2-3^(1/2),-2+3^(1/2)]",
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181 |
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182 | "roots(2*x^3 - 4x^2 - 22*x + 24)",
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183 | "[-3,1,4]",
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184 |
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185 | "clearall",
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186 | "",
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187 |
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188 |
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189 | "thePoly = 3*x^3 - 10*x^2 - 14*x + 27",
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190 | "",
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191 |
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192 | "roots(thePoly)",
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193 | "[1/3*(10/3-226/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))-(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)),1/3*(10/3+113/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)-113*i*3^(1/2)/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*i*3^(1/2)*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)),1/3*(10/3+113/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))+1/2*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3)+113*i*3^(1/2)/(9*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))-1/2*i*3^(1/2)*(781/54+i*97^(1/2)*1933^(1/2)/(2*3^(1/2)))^(1/3))]",
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194 |
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195 | "and((subst(float(last[1]),x,thePoly) == 0),(subst(float(last[2]),x,thePoly) == 0),(subst(float(last[3]),x,thePoly)) == 0)",
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196 | "1",
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197 |
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198 |
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199 | "clearall",
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200 | "",
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201 |
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202 |
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203 | "thePoly = 1*x^3 + 6*x^2 - 12*x + 8",
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204 | "",
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205 |
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206 | "roots(thePoly)",
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207 | "[-2+2^(1/3)+2^(2/3)-i*2^(1/3)*3^(1/2)+i*2^(2/3)*3^(1/2),-2+2^(1/3)+2^(2/3)+i*2^(1/3)*3^(1/2)-i*2^(2/3)*3^(1/2),-2*(1+2^(1/3)+2^(2/3))]",
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208 |
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209 | "and((subst(float(last[1]),x,thePoly) == 0),(subst(float(last[2]),x,thePoly) == 0),(subst(float(last[3]),x,thePoly) == 0))",
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210 | "1",
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211 |
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212 |
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213 | "roots(1*x^3 + 6*x^2 + 12*x + 8)",
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214 | "-2",
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215 |
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216 | "clearall",
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217 | "",
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218 |
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219 |
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220 | "thePoly = 1*x^3 + 0*x^2 + 12*x - 10",
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221 | "",
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222 |
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223 | "roots(thePoly)",
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224 | "[(-6+1/6*(-135+27*89^(1/2))^(2/3)-6*i*3^(1/2)-1/6*i*3^(1/2)*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3)),(-6+1/6*(-135+27*89^(1/2))^(2/3)+6*i*3^(1/2)+1/6*i*3^(1/2)*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3)),(12-1/3*(-135+27*89^(1/2))^(2/3))/((-135+27*89^(1/2))^(1/3))]",
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225 |
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226 | "and((subst(float(last[1]),x,thePoly) < float(10^(-13))),(subst(float(last[2]),x,thePoly) < float(10^(-13))),(subst(float(last[3]),x,thePoly) < float(10^(-14))))",
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227 | "1",
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228 |
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229 | "roots(1*x^3 + 0*x^2 - 18*x + 35)",
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230 | "[-5,5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]",
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231 |
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232 | "clearall",
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233 | "",
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234 |
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235 |
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236 | "thePoly = 1*x^3 + 0*x^2 - 3*x - 6",
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237 | "",
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238 |
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239 | "roots(thePoly)",
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240 | "[(3+1/3*(-81+54*2^(1/2))^(2/3)-3*i*3^(1/2)+1/3*i*3^(1/2)*(-81+54*2^(1/2))^(2/3))/(2*(-81+54*2^(1/2))^(1/3)),(3+1/3*(-81+54*2^(1/2))^(2/3)+3*i*3^(1/2)-1/3*i*3^(1/2)*(-81+54*2^(1/2))^(2/3))/(2*(-81+54*2^(1/2))^(1/3)),(-3-1/3*(-81+54*2^(1/2))^(2/3))/((-81+54*2^(1/2))^(1/3))]",
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241 |
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242 | "and((subst(float(last[1]),x,thePoly) < float(10^(-14))),(subst(float(last[2]),x,thePoly) < float(10^(-14))),(subst(float(last[3]),x,thePoly) < float(10^(-14))))",
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243 | "1",
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244 |
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245 | "roots(2*x^3 - 30*x^2 + 162*x - 350)",
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246 | "[7,4-3*i,4+3*i]",
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247 |
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248 | "roots(1*x^3 - 4*x^2 - 6*x + 5)",
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249 | "[5,-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2)]",
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250 |
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251 | "clearall",
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252 | "",
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253 |
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254 |
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255 | "thePoly = 3*x^3 + 6*x^2 + 4*x + 9",
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256 | "",
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257 |
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258 | "roots(thePoly)",
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259 | "[1/3*(-2-73^(1/3)),1/3*(-2+1/2*73^(1/3)-1/2*i*3^(1/2)*73^(1/3)),1/3*(-2+1/2*73^(1/3)+1/2*i*3^(1/2)*73^(1/3))]",
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260 |
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261 | "and((subst(float(last[1]),x,thePoly) < float(10^(-14))),(subst(float(last[2]),x,thePoly) < float(10^(-14))),(subst(float(last[3]),x,thePoly) < float(10^(-14))))",
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262 | "1",
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263 |
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264 | "clearall",
|
265 | "",
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266 |
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267 |
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268 | "thePoly = 3*x^3 + 21*x^2 + 2*x + 3",
|
269 | "",
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270 |
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271 | "roots(thePoly)",
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272 | "[1/3*(-7-47/((671/2+1/2*34949^(1/2))^(1/3))-(671/2+1/2*34949^(1/2))^(1/3)),1/3*(-7+47/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*(671/2+1/2*34949^(1/2))^(1/3)-47*i*3^(1/2)/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*i*3^(1/2)*(671/2+1/2*34949^(1/2))^(1/3)),1/3*(-7+47/(2*(671/2+1/2*34949^(1/2))^(1/3))+1/2*(671/2+1/2*34949^(1/2))^(1/3)+47*i*3^(1/2)/(2*(671/2+1/2*34949^(1/2))^(1/3))-1/2*i*3^(1/2)*(671/2+1/2*34949^(1/2))^(1/3))]",
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273 |
|
274 | "and((subst(float(last[1]),x,thePoly) < float(10^(-12))),(subst(float(last[2]),x,thePoly) < float(10^(-13))),(subst(float(last[3]),x,thePoly) < float(10^(-13))))",
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275 | "1",
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276 |
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277 | "clearall",
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278 | "",
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279 |
|
280 |
|
281 | "thePoly = 3*x^3 - 6*x^2 + 4*x - 5",
|
282 | "",
|
283 |
|
284 | "roots(thePoly)",
|
285 |
|
286 |
|
287 |
|
288 |
|
289 | "[2/3-1/3*(-1)^(1/3)*37^(1/3),1/3*(2-1/2*37^(1/3)+1/2*i*3^(1/2)*37^(1/3)),1/3*(2+37^(1/3))]",
|
290 |
|
291 | "and((subst(float(last[1]),x,thePoly) < float(10^(-14))),(subst(float(last[2]),x,thePoly) < float(10^(-14))),(subst(float(last[3]),x,thePoly) < float(10^(-14))))",
|
292 | "1",
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293 |
|
294 | "roots(8*x^3 - 36*x^2 + 54*x - 27)",
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295 | "3/2",
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296 |
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297 | "roots(3*x^3 - 5*x^2 - 1*x - 2)",
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298 | "[2,-1/6-1/6*i*11^(1/2),-1/6+1/6*i*11^(1/2)]",
|
299 |
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300 |
|
301 | "clearall",
|
302 | "",
|
303 |
|
304 |
|
305 | "thePoly = 3*x^3 - 6*x^2 + 4*x - i",
|
306 | "",
|
307 |
|
308 | "roots(thePoly)",
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309 | "[1/3*(2-(8-9*i)^(1/3)),1/3*(2+1/2*(8-9*i)^(1/3)-1/2*i*3^(1/2)*(8-9*i)^(1/3)),1/3*(2+1/2*(8-9*i)^(1/3)+1/2*i*3^(1/2)*(8-9*i)^(1/3))]",
|
310 |
|
311 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))",
|
312 | "1",
|
313 |
|
314 |
|
315 | "clearall",
|
316 | "",
|
317 |
|
318 |
|
319 | "thePoly = x^3+i",
|
320 | "",
|
321 |
|
322 | "roots(thePoly)",
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323 |
|
324 |
|
325 |
|
326 |
|
327 | "[-1/2*i-1/2*3^(1/2),-1/2*i+1/2*3^(1/2),i]",
|
328 |
|
329 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))",
|
330 | "1",
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331 |
|
332 |
|
333 | "clearall",
|
334 | "",
|
335 |
|
336 |
|
337 | "thePoly = x^3-i",
|
338 | "",
|
339 |
|
340 | "roots(thePoly)",
|
341 |
|
342 |
|
343 | "[-3/4*i-1/2*(-1)^(5/6)-1/4*3^(1/2),3/4*i-1/2*(-1)^(5/6)+1/4*3^(1/2),(-1)^(5/6)]",
|
344 |
|
345 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))))",
|
346 | "1",
|
347 |
|
348 |
|
349 |
|
350 | "clearall",
|
351 | "",
|
352 |
|
353 | "thePoly = x^4 + 1",
|
354 | "",
|
355 |
|
356 | "theRoots = roots(thePoly)",
|
357 | "",
|
358 |
|
359 | "theRoots",
|
360 |
|
361 | "[-1/2*2^(1/2)-1/2*i*2^(1/2),-1/2*2^(1/2)+1/2*i*2^(1/2),1/2*2^(1/2)-1/2*i*2^(1/2),1/2*2^(1/2)+1/2*i*2^(1/2)]",
|
362 |
|
363 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))",
|
364 | "1",
|
365 |
|
366 | "clearall",
|
367 | "",
|
368 |
|
369 | "thePoly = 4*x^4 - 1*x^3 + 4*x^2 + 3*x + 5",
|
370 | "",
|
371 |
|
372 | "theRoots = roots(thePoly)",
|
373 | "",
|
374 |
|
375 | "theRoots",
|
376 | "[1/16-1/2*(-125/96-447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)+1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16-1/2*(-125/96+447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)-1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16+1/2*(-125/96-447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)+1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2),1/16+1/2*(-125/96+447/(256*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2))-265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)-265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)-1/2*(-125/192+265/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))+1/3*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3)+265*i*3^(1/2)/(192*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))-1/3*i*3^(1/2)*(4417/1024+9/1024*i*461^(1/2)*1471^(1/2))^(1/3))^(1/2)]",
|
377 |
|
378 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
379 | "1",
|
380 |
|
381 | "clearall",
|
382 | "",
|
383 |
|
384 | "thePoly = x^5 + 1",
|
385 | "",
|
386 |
|
387 | "theRoots = roots(thePoly)",
|
388 | "",
|
389 |
|
390 | "theRoots",
|
391 |
|
392 | "[-1,cos(1/5*pi)+i*sin(1/5*pi),cos(3/5*pi)+i*sin(3/5*pi),-cos(2/5*pi)-i*sin(2/5*pi),-cos(4/5*pi)-i*sin(4/5*pi)]",
|
393 |
|
394 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[5]),x,thePoly))) < float(2*10^(-12))))",
|
395 | "1",
|
396 |
|
397 | "clearall",
|
398 | "",
|
399 |
|
400 | "thePoly = a*x^5 + k",
|
401 | "",
|
402 |
|
403 | "theRoots = roots(thePoly)",
|
404 | "",
|
405 |
|
406 | "theRoots[1] = simplify(theRoots[1])",
|
407 | "",
|
408 |
|
409 | "theRoots[1]",
|
410 | "-(-1)^(2/5)*((k/a)^(2/5))^(1/2)",
|
411 |
|
412 | "theRoots[2] = simplify(theRoots[2])",
|
413 | "",
|
414 |
|
415 | "theRoots[2]",
|
416 | "-(-1)^(4/5)*((k/a)^(2/5))^(1/2)",
|
417 |
|
418 | "theRoots[3] = circexp(theRoots[3])",
|
419 | "",
|
420 |
|
421 | "theRoots[3]",
|
422 | "exp(1/5*i*pi)*(k/a)^(1/5)",
|
423 |
|
424 | "theRoots[4] = circexp(theRoots[4])",
|
425 | "",
|
426 |
|
427 | "theRoots[4]",
|
428 | "exp(3/5*i*pi)*(k/a)^(1/5)",
|
429 |
|
430 | "theRoots[5] = simplify(theRoots[5])",
|
431 | "",
|
432 |
|
433 | "theRoots[5]",
|
434 | "-(k/a)^(1/5)",
|
435 |
|
436 |
|
437 |
|
438 |
|
439 |
|
440 |
|
441 |
|
442 |
|
443 |
|
444 |
|
445 |
|
446 | "clearall",
|
447 | "",
|
448 |
|
449 | "thePoly = x^3 - 7*x^2 + 41*x - 87",
|
450 | "",
|
451 |
|
452 | "theRoots = roots(thePoly)",
|
453 | "",
|
454 |
|
455 | "theRoots",
|
456 | "[3,2-5*i,2+5*i]",
|
457 |
|
458 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))",
|
459 | "1",
|
460 |
|
461 | "clearall",
|
462 | "",
|
463 |
|
464 | "thePoly = 398683376+1720835*x+2320*x^2+x^3",
|
465 | "",
|
466 |
|
467 | "theRoots = roots(thePoly)",
|
468 | "",
|
469 |
|
470 |
|
471 |
|
472 | "theRoots",
|
473 | "[-2320/3-219895/(3*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))-1/3*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3),-2320/3+219895/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)-219895*i*3^(1/2)/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*i*3^(1/2)*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3),-2320/3+219895/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))+1/6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)+219895*i*3^(1/2)/(6*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3))-1/6*i*3^(1/2)*(-96123824+9*i*7^(1/2)*79^(1/2)*2297^(1/2)*13538519^(1/2))^(1/3)]",
|
474 |
|
475 |
|
476 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-7))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-7))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-7))))",
|
477 | "1",
|
478 |
|
479 | "clearall",
|
480 | "",
|
481 |
|
482 | "thePoly = x^4 - 1*x^3 + 4*x^2 + 3*x + 5",
|
483 | "",
|
484 |
|
485 | "theRoots = roots(thePoly)",
|
486 | "",
|
487 |
|
488 | "theRoots",
|
489 | "[-1/2-1/2*i*3^(1/2),-1/2+1/2*i*3^(1/2),1-2*i,1+2*i]",
|
490 |
|
491 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
492 | "1",
|
493 |
|
494 | "clearall",
|
495 | "",
|
496 |
|
497 | "thePoly = x^4 - 2*x^3 - 7*x^2 + 8*x + 12",
|
498 | "",
|
499 |
|
500 | "theRoots = roots(thePoly)",
|
501 | "",
|
502 |
|
503 | "theRoots",
|
504 | "[-2,-1,2,3]",
|
505 |
|
506 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
507 | "1",
|
508 |
|
509 | "clearall",
|
510 | "",
|
511 |
|
512 | "thePoly = x^4+8*x^2+3",
|
513 | "",
|
514 |
|
515 | "theRoots = roots(thePoly)",
|
516 | "",
|
517 |
|
518 | "theRoots",
|
519 | "[-(-4-13^(1/2))^(1/2),-(-4+13^(1/2))^(1/2),(-4-13^(1/2))^(1/2),(-4+13^(1/2))^(1/2)]",
|
520 |
|
521 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
522 | "1",
|
523 |
|
524 | "clearall",
|
525 | "",
|
526 |
|
527 | "thePoly = -1*x^3-1*x^2+10*x - 8",
|
528 | "",
|
529 |
|
530 | "theRoots = roots(thePoly)",
|
531 | "",
|
532 |
|
533 | "theRoots",
|
534 | "[-4,1,2]",
|
535 |
|
536 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))",
|
537 | "1",
|
538 |
|
539 | "clearall",
|
540 | "",
|
541 |
|
542 | "thePoly = -3-9*x+3*x^2+x^3",
|
543 | "",
|
544 |
|
545 | "theRoots = roots(thePoly)",
|
546 | "",
|
547 |
|
548 |
|
549 |
|
550 |
|
551 |
|
552 |
|
553 |
|
554 |
|
555 | "theRoots",
|
556 |
|
557 | "[-1+cos(1/9*pi)-cos(8/9*pi)+i*sin(1/9*pi)-i*sin(8/9*pi)-2*3^(1/2)*cos(11/18*pi),-1+cos(1/9*pi)-cos(8/9*pi)+i*sin(1/9*pi)-i*sin(8/9*pi)+2*3^(1/2)*cos(11/18*pi),-1-2*cos(1/9*pi)+2*cos(8/9*pi)-2*i*sin(1/9*pi)+2*i*sin(8/9*pi)]",
|
558 |
|
559 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))",
|
560 | "1",
|
561 |
|
562 | "clearall",
|
563 | "",
|
564 |
|
565 | "thePoly = x^4 + 8*x^3 + 12*x^2 + (2*30^(1/2) -16)*x + 4*30^(1/2)-28",
|
566 | "",
|
567 |
|
568 | "theRoots = roots(thePoly)",
|
569 | "",
|
570 |
|
571 | "theRoots",
|
572 | "[-2-1/2*(18-4*5^(1/2))^(1/2)+3^(1/2)/(2^(1/2)),-2-1/2*(18+4*5^(1/2))^(1/2)-1/2*2^(1/2)*3^(1/2),-2+1/2*(18-4*5^(1/2))^(1/2)+3^(1/2)/(2^(1/2)),-2+1/2*(18+4*5^(1/2))^(1/2)-1/2*2^(1/2)*3^(1/2)]",
|
573 |
|
574 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
575 | "1",
|
576 |
|
577 | "clearall",
|
578 | "",
|
579 |
|
580 | "thePoly = x^3 + x - 2",
|
581 | "",
|
582 |
|
583 | "theRoots = roots(thePoly)",
|
584 | "",
|
585 |
|
586 | "theRoots",
|
587 | "[1,-1/2-1/2*i*7^(1/2),-1/2+1/2*i*7^(1/2)]",
|
588 |
|
589 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))",
|
590 | "1",
|
591 |
|
592 |
|
593 | "clearall",
|
594 | "",
|
595 |
|
596 | "thePoly = x^3 + x^2 - 7",
|
597 | "",
|
598 |
|
599 | "theRoots = roots(thePoly)",
|
600 | "",
|
601 |
|
602 |
|
603 | "and(abs(float(subst(theRoots[1],x,thePoly))) < float(2*10^(-12)),abs(float(subst(theRoots[2],x,thePoly))) < float(2*10^(-12)),abs(float(subst(theRoots[3],x,thePoly))) < float(2*10^(-12)))",
|
604 | "1",
|
605 |
|
606 |
|
607 |
|
608 | "clearall",
|
609 | "",
|
610 |
|
611 | "thePoly = x^4 + 8*x^2 + 3",
|
612 | "",
|
613 |
|
614 | "theRoots = roots(thePoly)",
|
615 | "",
|
616 |
|
617 | "theRoots",
|
618 | "[-(-4-13^(1/2))^(1/2),-(-4+13^(1/2))^(1/2),(-4-13^(1/2))^(1/2),(-4+13^(1/2))^(1/2)]",
|
619 |
|
620 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(8*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(8*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))",
|
621 | "1",
|
622 |
|
623 | "clearall",
|
624 | "",
|
625 |
|
626 | "thePoly = x^4 - 10*x^3 + 21*x^2 + 40*x - 100",
|
627 | "",
|
628 |
|
629 | "theRoots = roots(thePoly)",
|
630 | "",
|
631 |
|
632 | "theRoots",
|
633 | "[-2,2,5]",
|
634 |
|
635 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))))",
|
636 | "1",
|
637 |
|
638 | "clearall",
|
639 | "",
|
640 |
|
641 | "thePoly = 2*x^4 - 8*x^3 + 2*x^2 + 24*x - 14",
|
642 | "",
|
643 |
|
644 | "theRoots = roots(thePoly)",
|
645 | "",
|
646 |
|
647 | "clearall",
|
648 | "",
|
649 |
|
650 | "thePoly = x^4 - 4*x^3 + x^2 + 12*x - 7",
|
651 | "",
|
652 |
|
653 | "theRoots = roots(thePoly)",
|
654 | "",
|
655 |
|
656 | "theRoots",
|
657 | "[-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2),5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]",
|
658 |
|
659 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
660 | "1",
|
661 |
|
662 | "clearall",
|
663 | "",
|
664 |
|
665 | "thePoly = 2*x^4 - 8*x^3 + 2*x^2 + 24*x - 14",
|
666 | "",
|
667 |
|
668 | "theRoots = roots(thePoly)",
|
669 | "",
|
670 |
|
671 | "theRoots",
|
672 | "[-1/2-1/2*5^(1/2),-1/2+1/2*5^(1/2),5/2-1/2*i*3^(1/2),5/2+1/2*i*3^(1/2)]",
|
673 |
|
674 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
675 | "1",
|
676 |
|
677 | "clearall",
|
678 | "",
|
679 |
|
680 | "thePoly = x^4 - 9*x^3 + 22*x^2 + 28*x - 120",
|
681 | "",
|
682 |
|
683 | "theRoots = roots(thePoly)",
|
684 | "",
|
685 |
|
686 | "theRoots",
|
687 | "[-2,3,4-2*i,4+2*i]",
|
688 |
|
689 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-12))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-12))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-12))))",
|
690 | "1",
|
691 |
|
692 |
|
693 |
|
694 |
|
695 |
|
696 |
|
697 |
|
698 |
|
699 |
|
700 |
|
701 |
|
702 |
|
703 |
|
704 |
|
705 |
|
706 |
|
707 |
|
708 |
|
709 |
|
710 |
|
711 |
|
712 |
|
713 |
|
714 | "clearall",
|
715 | "",
|
716 |
|
717 | "thePoly = x^4 + 2*x^3 - 41*x^2 - 42*x + 360",
|
718 | "",
|
719 |
|
720 | "theRoots = roots(thePoly)",
|
721 | "",
|
722 |
|
723 | "theRoots",
|
724 | "[-6,-4,3,5]",
|
725 |
|
726 | "and((abs(float(subst(float(last[1]),x,thePoly))) < float(2*10^(-15))),(abs(float(subst(float(last[2]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[3]),x,thePoly))) < float(2*10^(-15))), (abs(float(subst(float(last[4]),x,thePoly))) < float(2*10^(-15))))",
|
727 | "1",
|
728 |
|
729 |
|
730 |
|
731 |
|
732 | "thePoly = quote(thePoly)",
|
733 | "",
|
734 |
|
735 | "theRoots = quote(theRoots)",
|
736 | "",
|
737 |
|
738 | ]
|