Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-02T20:30:22.339Z Has data issue: false hasContentIssue false

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)