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Descartes’ Rule of Signs. To accompany TMA10
Lecture Notes: For more info about the Descartes’ Rule of signs, or any topic in algebra for that matter, Google it. For Descartes’ Rule of signs, follow this link: http://www.purplemath.com/modules/drofsign.htm • Use Descartes' Rule of Signs to determine possible combinations of positive, negative and imaginary roots for f (x) = x5 – x4 + 3x3 + 9x2 – x + 5. • (Pag-pronounce ng Descartes, silent ang “s” kaya “dekart” ang bigkas nito “dekart’s” naman ang bigkas ng “Descartes’ “) Ginagamit ang Descartes’ Rule of Signs SA pagbilang ng posibleng combo ng positive roots at negative roots at SA gayon ng imaginary roots din. Paano? Ganito yon: • Bilang ng positive roots.

Rule: From the original equation f(x) = 0, count the number of variations of signs in f(x) from + to – or from – to +. The possible number of positive roots is this number of variations or less by an even cardinal number or counting number such that the difference is a cardinal number too.

Mula sa signs ng orihinal na equation na f (x) = x5 – x4 + 3x3 + 9x2 – x + 5 |Term |x5 |– x4 |+ 3x3 |+ 9x2 |– x |+ 5 | |sign |+ |– |+ |+ |– |+ |

Bilangin ang pagbabago ng mga signs ng bawat term ng orihinal Na equation mula + to – sign o mula – to + sign. Sa halimbawa na f (x) = x5 – x4 + 3x3 + 9x2 – x + 5 ganito ang sequence ng mga signs ng orihinal na equation: + - + + - + at ang bilang ng pagbabago ng signs ay 4. Ang mga posibleng bilang ng positive roots ay 4 o bilang ng variation or less by an even cardinal number such that the resulting difference is a cardinal number too. Kaya ang posibleng bilang ng positive roots ay 4, 4-2, 4-4 o 4, 2, 0. Sa table form, eto yong posibleng bilang ng positive roots. |posibleng bilang ng positive roots |4 |2 |0 |

• Bilang ng negative roots.
Rule: From the original equation f(x) = 0, find...
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