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CBSE Class 10 Answered

if alpha and beta are the zeros of the polynomial ax2+bx+c find the value of alpha3+beta3
Asked by mehraharshit.dk000 | 05 Oct, 2023, 08:05: PM
answered-by-expert Expert Answer
 
begin mathsize 14px style a space x squared space plus space b space x space plus space c space equals space 0 end style
If roots of above quatratic equations are α and β , then
 
begin mathsize 14px style alpha space equals space fraction numerator negative b plus square root of b squared minus 4 a c end root over denominator 2 a end fraction space space space a n d space space beta space equals space fraction numerator negative b minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end style
begin mathsize 14px style alpha cubed space plus space beta cubed space equals space left parenthesis space alpha space plus space beta space right parenthesis space left parenthesis space alpha squared space minus space alpha beta space plus space beta squared space right parenthesis end style ..................................(1)
begin mathsize 14px style alpha plus beta space equals negative space b over a end style   ................................. (2)
begin mathsize 14px style alpha squared plus beta squared space equals space fraction numerator 1 over denominator 4 a squared end fraction open square brackets left parenthesis negative b plus square root of b squared minus 4 a c end root space right parenthesis squared plus left parenthesis negative b minus square root of b squared minus 4 a c end root space right parenthesis squared space close square brackets space end style
begin mathsize 14px style alpha squared plus beta squared space equals space fraction numerator 1 over denominator 4 a squared end fraction open square brackets 2 b squared plus 2 left parenthesis b squared minus 4 a c right parenthesis close square brackets space equals space fraction numerator left parenthesis b squared minus a c right parenthesis over denominator a squared end fraction end style
begin mathsize 14px style alpha space beta space equals space fraction numerator 1 over denominator 4 a squared end fraction open parentheses b squared space minus space b squared plus 4 a c close parentheses space equals space c over a end style
begin mathsize 14px style alpha squared space minus alpha beta space plus beta squared space equals space fraction numerator left parenthesis b squared minus a c right parenthesis over denominator a squared end fraction minus fraction numerator a c over denominator a squared end fraction space equals space fraction numerator b squared minus 2 a c over denominator a squared end fraction end style ......................... (3)
By using eqn.(2) and eqn.(3) , we rewrite eqn.(1) as
 
begin mathsize 14px style alpha cubed space plus space beta cubed space equals space fraction numerator left parenthesis 2 a b c minus b cubed right parenthesis over denominator a cubed end fraction end style
 
Answered by Thiyagarajan K | 05 Oct, 2023, 10:31: PM

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