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:

x³ - 4x² + 5x - 2; 2, 1, 1

Solution

माना p(x) = x³ - 4x² + 5x - 2
दिए हुए बहुपद की ax³ + bx² + cx + d से तुलना करने पर, हम प्राप्त करते है:
a = 1, b = - 4, c = 5 और d = - 2
अब,
$straight p left parenthesis space 2 space right parenthesis space equals space left parenthesis space 2 space right parenthesis cubed space minus space 4 space left parenthesis space 2 space right parenthesis squared space plus space 5 space cross times space left parenthesis space 2 space right parenthesis space minus space 2 space space space space space space space space space equals space 8 space minus space 16 space plus space 10 space minus space 2 space equals space 18 space minus space 18 space equals space 0$
और
$straight P left parenthesis space 1 space right parenthesis space equals space left parenthesis space 1 space right parenthesis cubed space minus space 4 space cross times space left parenthesis space 1 space right parenthesis squared space plus space 5 space cross times space 1 space minus space 2 space space space space space space space space space equals space 1 space minus space 4 space plus space 5 space minus space 2 space equals space 6 space minus space 6 space equals space 0$
∴ 2, 1, 1 बहुपद ( x³ - 4x² + 5x - 2 ) के शून्यक है
अत: सत्यापित हुआ

$straight alpha space equals space 2 comma space straight beta space equals space 1 space और space straight gamma space equals space 1$
$straight alpha space plus space straight beta space plus space straight gamma space equals space 2 space plus space 1 space plus space 1 space equals space 4 space equals space fraction numerator negative space left parenthesis space minus space 4 space right parenthesis over denominator 1 end fraction space equals space fraction numerator negative space straight b over denominator straight a end fraction$
$αβ space plus space βγ space plus space γα space equals space 2 space cross times space 1 space plus space 1 space cross times space 1 space plus space 1 space cross times space 2$
$equals space 2 space plus space 1 space plus space 2 space equals space 5 over 1 space equals space straight c over straight a$
और $αβγ space equals space 2 space cross times space 1 space cross times space 1 space equals space 2 space equals space fraction numerator negative space left parenthesis space minus space 2 space right parenthesis over denominator 1 end fraction space equals space fraction numerator negative space straight d over denominator straight a end fraction$
अत: सत्यापित हुआl

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