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Notes on the Theory of Equations

Published online by Cambridge University Press:  03 November 2016

Extract

Multiply throughout by a2 and write y for ax+ b ; the equation becomes

where Hacb2, G ≡ a2d − 3abc + 2b3.

Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written

Accordingly the two remaining roots are also real if

But since α satisfies (i),

and so

Hence if (i) has three real roots, G2 +4H3 ≤ 0; and clearly, when G2+4H3 = 0, two roots are numerically equal to and the third to .

Type
Research Article
Copyright
Copyright © Mathematical Association 1939 

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