If x^5-1 has roots 1,a,b,c,d find the value of (1-a)(1-b)(1-c)(1-d). Experts please help... Share with your friends Share 1 Neha Sethi answered this Dear student Since 1,a,b,c,d are the roots of x5-1So, x-1, x-a, x-b, x-c, x-d are the factors of x5-1.i.e x5-1=x-1x-ax-bx-cx-d=x2-ax-x+ax2-cx-bx+bcx-d=x4-cx3-bx3+bcx2-ax3+acx2+abx2-abcx-x3+cx2+bx2-bcx+ax2-acx-abx+abcx-d=x4-a+b+c+1x3+bc+ac+ab+a+b+cx2-abc+ab+bc+acx+abcx-d=x5-dx4-a+b+c+1x4+a+b+c+1dx3+bc+ac+ab+a+b+cx3-bc+ac+ab+a+b+cdx2-abc+ab+bc+acx2+abc+ab+bc+acdx+abcx-abcd=x5-a+b+c+d+1x4+ad+bd+cd+d+bc+ac+ab+a+b+cx3-bcd+acd+abd+ad+bd+dc+abc+ab+bc+acx2+abcd+abd+bcd+acd+abcx-abcdOn comparing the coeff. we geta+b+c+d+1=0 ....1ad+bd+cd+d+bc+ac+ab+a+b+c=0 ...2bcd+acd+abd+ad+bd+dc+abc+ab+bc+ac=0 ...3abcd+abd+bcd+acd+abc=0 ...4abcd=1 ....5Putting the value of abcd in 4, we getabd+bcd+acd+abc=-1Putting the above value in 3, we getad+bd+dc+ab+bc+ac=1Putting the above value in 2, we getd+a+b+c=-1Consider 1-a1-b1-c1-d=1-b-a+ab1-d-c+cd=1-d-c+cd-b+bd+bc-bcd-a+ad+ac-acd+ab-abd-abc+abcd=1-a+b+c+d+cd+bd+bc+ad+ac+ab-bcd+acd+abd+abc+abcd=1--1+1--1+1=1+1+1+1+1=5(∵ abcd=1 ,abd+bcd+acd+abc=-1,ad+bd+dc+ab+bc+ac=1,d+a+b+c=-1) Regards 1 View Full Answer Sannidhya Mate answered this can no one answer 0
