document.write( "Question 31432: Assume that a and b are different zeros of p(x)= x^3+kx-1. Show that ab is a zero of q(x)=x^3-kx-1 where k is any real number. \n" ); document.write( "

Algebra.Com's Answer #18253 by venugopalramana(3286) $\"\"$ $\"About$

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X^3+KX-1=0.....A AND B ARE 2 ROOTS.SINCE THIS IS A CUBIC IN X ,THERE
\n" ); document.write( "WILL BE 3 ROTS .LET THE THIRD ROOT BE C.SO WE KNOW THAT...
\n" ); document.write( "FOR X^3+0*X^2+KX-1=0
\n" ); document.write( "S1=A+B+C=0....OR...C= -(A+B)....................I
\n" ); document.write( "S2=AB+BC+CA=K.....OR.....AB+C(A+B)=K..............II
\n" ); document.write( "S3=ABC=1.....OR......C=1/AB...........................................................III
\n" ); document.write( "SUBSTITUTING -C FOR (A+B) IN EQN.II FROM EQN.I WE GET
\n" ); document.write( "AB-C*C=K.....SUBSTITUTING FOR C FROM EQN.III WE GET....
\n" ); document.write( "AB-1/(AB)^2-K=0.............MULTIPLYING BY (AB)^2..WE GET
\n" ); document.write( "(AB)^3-1-K(AB)^2=0........................................................IV
\n" ); document.write( "PUTTING X IN PLACE OF AB WE GET
\n" ); document.write( "X^3-KX^2-1=0.......IS THE EQN. SATISFIED BY AB IN VIEW OF WHAT WE
\n" ); document.write( "PROVED IN EQN.IV \n" ); document.write( "

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