XIV CONTENTS.
CHAPTER PAGE
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XLVII. Determinants 435 Minors 437 Vanishing of a Determinant 438 Multiplication of a Determinant 438 A Determinant expressed as the Sum of Two Other Determinants 439 Simplification of Determinants 440 Solution of Simultaneous Equations of the First Degree 444 Signs of the Terms 447 Determinant of Lower Order 448
XLVIII. Theory of Equations 450 Horner's Method of Synthetic Division 451 Number of the Roots 453 Depression of Equations 453 Formation of Equations 454 Relations between the Roots and the Coefficients 454 Fractional Roots 458 Imaginary Roots 458 Transformation of Equations 460 Standard Form of Reciprocal Equations 466 Descartes' Rule of Signs 469 Derived Functions 472 Equal Roots 472 Location of the Roots 474 Sturm's Theorem and Method 476 Graphical Representation of Functions 480 Solution of Higher Numerical Equations 483 Newton's Method 483 The Cube Roots of Unity 486 Cardan's Method 487 Biquadratic Equations 488 Incommensurable Roots 491 Horner's Method of Approximation 492 Any Root of Any Number 495 Miscellaneous Examples VII 496
