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4.1. Simultaneous Equations in Two or Three Unknowns^123 (b) Prove that the equations have a root in common if and only if (cp - ...
124 4. Equations (4 x+y+z = 15 x2+y2 +z2 = 83 x3 + y3 + z3 = 495 (b) x+y+z = 2 x2 +y2 + z2 = 14 xyz = -6. By factoring the left ...
4.2. Surd Equations 125 Exercises Consider the equation &??? + m = 1. (a) As part of a strategy to obtain an equation with ...
126 4. Equations (a) Verify that, if w = $&+ fi, then (b) Solve the equation Solve the equation x2+18x+30=2 x2+18x+45. ...
4.3. Solving Special Polynomial Equations 127 Theorem of Algebra that every complex polynomial had a zero by actually expressing ...
128 4. Equations (c) Use 2 cos n0 = (cos 0 + i sin 0)” + (COS 0 - i sin t9)n to verify that 2cos2e=(2cose)y-2=y2-2 2cos3e = (2co ...
4.3. Solving Special Polynomial Equations 129 We might try to factor it as the product of a quadratic and a cubic: (t2 + ut + v) ...
130 4. Equations so it makes sense to denote this extension by Q(fi, fi) and to regard it as the field obtained from Q by adjoin ...
4.3. Solving Special Polynomial Equations 131 (a) Solve the equation f(t) =^0 by Cardan’s Method (Exercise 1.4.4). To find u3 an ...
132 4. Equations and quartics of finding the roots of polynomial equations of arbitrary de- gree. To formulate the type of solut ...
4.3. Solving Special Polynomial Equations 133 The polynomial t4 - 4t3 + 6t2 - 4t - 1 = (t - 1)4 - 2 is irreducible over Q, and t ...
134 4. Equations Thus we can write (123). (1243) = (1432). Unlike a product of numbers, this product is not commutative; sometim ...
4.3. Solving Special Polynomial Equations 135 similar way, eliminate (132), (142), (1342), (1432), (13), (134), (234), (14), (23 ...
136 4. Equations express the relationship between the groups G, Gi and HI by the notation: G1 Y G/HI. Finally, we subject the ba ...
4.3. Solving Special Polynomial Equations 137 (1) given an arbitrary angle, to construct an angle of one third the mag- nitude u ...
138 4. Equations 4.4 The Fundamental Theorem of Algebra: Intersecting Curves The Fundamental Theorem of Algebra states that ever ...
4.4. The Fundamental Theorem of Algebra 139 as a simultaneous real system ax2 + bx - ay2 + c = (^0) w y(2ax + b) = 0. CL) (b) Sh ...
140 4. Equations (a) z2 + 2% + 3 = 0 (b) .z2 + 22 - 3 = 0 (cl z2 -6z+9=0 (d) z2 + 2iz - 1 = 0 (e) z2 + (1 + 2i)z + (-1 + i) = 0. ...
4.4. The Fundamental Theorem of Algebra 141 Solve graphically the equations: (a) z3 - 72 + 6 = 0 (b) .z3 - t - 1 = 0 (c) 23 + ...
142 4. Equations v(r cos 0, r sin 0) = r” sin n0 + terms of lower degree in r. Since r is very large, the terms involving the hi ...
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