Question

सर्यपित कीजिए कि निम्न त्रिघात बहुपदों के साथ दी गई संख्याएं शून्यक हैं प्रत्येक स्थिति में शुन्यकों और गुणकों के बीच के सम्बन्ध को सत्यापित कीजिए:

2x³+ x²–5x + 2; $1 half$ , 1,–2

Solution

माना p(x) = 2x³+x²–5x + 2 दिए हुए बहुपद कि ax³ + bx³+ cx + d से तुलना करने पर, हमें प्राप्त होता है:
a = 2, b = 1, c = -5, d = 2
अब

straight p open parentheses space 1 half space close parentheses space equals space 2 space cross times space open parentheses space 1 half space close parentheses cubed space plus space open parentheses space 1 half space close parentheses squared space minus space 5 space cross times space open parentheses space 1 half space close parentheses space plus space 2

equals space 2 space cross times space 1 over 8 space plus space 1 fourth space minus space 5 over 2 space plus space 2

equals space 1 fourth space plus space 1 fourth space minus space 5 over 2 space plus space 2

equals space fraction numerator 1 space plus space 1 space minus space 10 space plus space 8 over denominator 4 end fraction space equals space 0 over 4

straight p left parenthesis 1 right parenthesis space equals space 2 space cross times space left parenthesis space 1 space right parenthesis cubed space plus space left parenthesis space 1 space right parenthesis squared space minus space 5 space cross times space 1 space plus space 2

equals space 2 space plus space 1 space minus space 5 space plus space 2 space equals space 0

और

straight p left parenthesis 2 right parenthesis space equals space 2 space left parenthesis space minus space 2 space right parenthesis cubed space plus space left parenthesis space minus space 2 space right parenthesis squared space minus space 5 space cross times space left parenthesis space minus space 2 space right parenthesis space plus space 2

equals space minus space 16 space plus space 4 space plus space 10 space plus space 2 space equals space 0

∴

1 half comma space 1 space और space left parenthesis space minus 2 space right parenthesis

बहुपद

2 straight x cubed space plus space straight x squared space minus space 5 straight x space plus space 2

के शून्यक हैl
अत: यह सत्यापित हुआ

straight alpha space equals space 1 half comma space straight beta space equals space 1 space और space straight gamma space equals space minus space 2

straight alpha space plus space straight beta space plus space straight gamma space equals space 1 half space plus space 1 space plus space left parenthesis space minus space 2 space right parenthesis space equals space minus space 1 half space equals space minus space straight b over straight a

αβ space plus space βγ space plus space γα space equals space 1 half space cross times space 1 space plus space 1 space cross times space left parenthesis space minus space 2 space right parenthesis space plus space left parenthesis space minus space 2 space right parenthesis space cross times space 1 half

equals space 1 half space minus space 2 space minus space 1 space equals space fraction numerator 1 space minus space 4 space minus space 2 over denominator 2 end fraction space equals space fraction numerator negative space 5 over denominator 2 end fraction space equals space straight c over straight a

और

αβγ space equals space 1 half space cross times space 1 space cross times space left parenthesis space minus space 2 space right parenthesis space space space space space space equals space fraction numerator negative 2 over denominator 2 end fraction equals fraction numerator negative straight d over denominator straight a end fraction

अत: सत्यापित हुआl

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