1.

if(3x ^ 2+x + 1 )/((x - 1 ) ^4)= (a)/((x- 1 )) +(b)/((x -1) ^ 2 )+(c )/((x-1)^3)+ (d)/((x-1)^ 4 ),then[{:(a,b),(c,d):}]

Answer»

`[{:(3,7),(5,0):}] `
`[{:(0,3),(7,5):}]`
`[{:(0,7),(3,5):}] `
`[{:(3,5),(7,0):}] `

SOLUTION : ` ( 3 x ^ 2+x+ 1) /(( x- 1 ) ^ 4 )=(a)/((x - 1 ))+(B)/(( x - 1 ) ^ 2)+ (c )/((x - 1 ) ^3 ) +(d)/((x - 1 ) ^4) `
`rArr3x ^ 2+ x+1=3 ( x - 1+ 1 ) ^ 2+(x - 1 )+2`
`= 3 (x - 1 ) ^ 2+6 ( x - 1 )+3 +(x - 1 )+ 2`
`=3 ( x - 1 ) ^ 2 +7 ( x - 1 ) + 5 `
`therefore(3x ^ 2+x + 1 )/( (x - 1 )^4 ) = (3 (x - 1 ) ^2 +7 ( x - 1 )+ 5 )/((x - 1 ) ^ 4 ) `
`= (3 ) /((x - 1 ) ^2 ) +(7 ) /((x - 1 ) ^(3)) +(5 ) /((x -1 ) ^ 4 )+(0)/((x - 1 )) `
`therefore(a ) /((x - 1 )) + (b)/((x - 1 )^ 2 )+(c )/((x -1 ) ^(3))+(d)/((x - 1 ) ^4 ) = (3 ) /((x - 1 ) ^2) `
`+(7 ) /((x -1) ^3 )+(5 ) /((x -1) ^4)+ (0)/((x - 1 )) `
`therefore [{:(a, b ),(c, d ):}] = [{:(0, 3 ) ,(7,5 ) :}] `


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