## THE THEORY OF EQUATIONS: WITH AN INTRODUCITON TO THE THEORY OF BINARY ALGEBRAIC FORMS |

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### Common terms and phrases

absolute term algebraic Analyse the equation applying biquadratic equation calculation changes of sign column commensurable root constituents corresponding cubic equation differences diminishing the roots divisors easily eliminating equa equal roots equation whose roots equation xi expressed in terms figure form the equation given equation gives hence homogeneous function Horner's Horner's method imaginary quantities imaginary roots integral function integral roots linear transformation method Miscellaneous Examples multiple roots multiplied negative root Newton's method nomographic nth degree nth root number of changes number of real numerical equations obtain pair polynomial positive root proceed Prop proposed equation proposition Prove quadratic factors quartic quotient real roots reciprocal reducing cubic result roots lie roots of unity solution Sturm's Sturm's theorem substituting subtracting suffixes symmetric function theorem tion tivo transformed equation trial-divisor vanish variables whence zero

### Popular passages

Page 175 - Series, f(x\ f'(x), ri(x), j"2(x), . . . , rn(x), the difference between the number of changes of sign in the series when a is substituted for x and...

Page 135 - The first method which suggests itself is one similar to that usually given to determine the coefficients of the equation whose roots are the squares of the differences of the roots of any given equation.

Page 28 - This rule, which enables us, by the mere inspection of a given equation, to assign a superior limit to the number of its positive roots, may be enunciated as follows : — No equation can have more positive roots than it has changes of sign from + to—, and from — to +, in the terms of its first member. We shall content ourselves for the present with the proof...

Page 190 - We have, therefore, a fraction in its lowest terms equal to an integer, which is impossibleHence j- cannot be a root of the equation. The real roots of the equation, therefore, are either integers or incommensurable quantities. Every equation whose coefficients are finite numbers, fractional or not, can be reduced to the form in which the coefficient of the first term is unity and those of the other terms whole numbers (Art. 31) ; so that in this way, by the aid of a simple transformation, the determination...

Page 36 - The coefficient p, of the fourth term with its sign changed is equal to the sum of the products of the roots taken three by three ; and so on, the signs of the coefficients...

Page 20 - Every equation of an odd degree has at least one real root ; and if there be but one, that root must necessarily have a contrary sign to that of the last term. - 4".

Page 226 - In the Propositions of the present and following Articles are contained the most important elementary properties of determinants which, by the aid of Cauchy's notation above described, render the employment of these functions of such practical advantage. PROP. I. — If any two rows, or any two columns, of a determinant be interchanged, the sign of the determinant is changed. This follows at once from the mode of formation (Rule (2), Art. 108), for an interchange of two rows is the same as an interchange...

Page 79 - The expression in brackets is called the discriminant of the cubic, and is represented by A ; giving the identities EXAMPLES.

Page 157 - If each negative coefficient be taken positively and divided by the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased by unity, is a superior limit of lhe positive roots. Let the equation be f(x) = 0, where f(x) denotes p,l'•K" + plx"~l+p,xn~13-pllx"~a + p,x"~í + ... -px"