Question 1038903
{{{(x^3 + 4x^2 + 20x - 7)/(( x - 1)^2( x^2 + 8))=a/(x-1)+b/(x-1)^2+(cx+d)/(x^2+8)}}}
{{{x^3 + 4x^2 + 20x - 7=a(x-1)(x^2+8)+b(x^2+8)+(cx+d)(x-1)^2}}}
{{{x^3 + 4x^2 + 20x - 7=a(x^3-x^2+8x-8)+b(x^2+8)+(cx+d)(x^2-2x+1)}}}
{{{x^3 + 4x^2 + 20x - 7=a(x^3-x^2+8x-8)+b(x^2+8)+c(x^3-2x^2+x)+d(x^2-2x+1)}}}
.
.
.
{{{x^3}}} terms,
{{{1=a+c}}}
.
.
{{{x^2}}} terms,
{{{4=-a+b-2c+d}}}
.
.
{{{x}}} terms,
{{{20=8a+c-2d}}}
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constant term,
{{{-7=-8a+8b+d}}}
Using Cramer's rule,
{{{A=(matrix(4,4,
1,0,1,0,
-1,1,-2,1,
8,0,1,-2,
-8,8,0,1))}}}
{{{abs(A)=81}}}
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{{{A[a]=(matrix(4,4,
1,0,1,0,
4,1,-2,1,
20,0,1,-2,
-7,8,0,1))}}}
{{{abs(A[a])=243}}}
.
.
{{{A[b]=(matrix(4,4,
1,1,1,0,
-1,4,-2,1,
8,20,1,-2,
-8,-7,0,1))}}}
{{{abs(A[b])=162}}}
.
.
{{{A[c]=(matrix(4,4,
1,0,1,0,
-1,1,4,1,
8,0,20,-2,
-8,8,-7,1))}}}
{{{abs(A[c])=-162}}}
.
.
{{{A[d]=(matrix(4,4,
1,0,1,1,
-1,1,-2,4,
8,0,1,20,
-8,8,0,-7))}}}
{{{abs(A[d])=81}}}
.
.
So,
{{{a=abs(A[a])/abs(A)=243/81=3}}}
{{{b=abs(A[b])/abs(A)=162/81=2}}}
{{{c=abs(A[c])/abs(A)-162/81=-2}}}
{{{d=abs(A[d])/abs(A)=81/81=1}}}
So,
{{{(x^3 + 4x^2 + 20x - 7)/(( x - 1)^2( x^2 + 8))=highlight(3/(x-1)+2/(x-1)^2+(1-2x)/(x^2+8))}}}