Find the partial fraction decomposition for the following rational expression. x3+x2+4x+7/(x2+1)2=f(x)/x2+1 +g(x)/(x2+1)2 Answer f(x)= and g(x)=

Question

Find the partial fraction decomposition for the following rational expression.

x³+x²+4x+7/(x²+1)²=f(x)/x²+1 +g(x)/(x²+1)²

Answer f(x)= and g(x)=

Answer 1

x³+x²+4x+7/(x²+1)²=f(x)/x²+1 +g(x)/(x²+1)²

Axl, repeated root like (x²+1) and (x²+1)² require solving simultaneous equations

(cx+d)/[(x²+1)² ]+(ax+b)/(x²+1)=x³+x²+4x+7/(x²+1)²

ax³+ax+bx²+b+cx+d=x³+x²+4x+7

a=1

a+c=4, c=3

b=1

b+d=7, d=6

(3x+6)/[(x²+1)² ]+(x+1)/(x²+1)=(x³+x²+4x+7)/((x²+1)²)

g(x)=3x+6, f(x)=x+1

Answer 2

For the sake of national clarity, A = f(x), B = g(x)

(x³ + x² + 4x + 7) / (x² + 1)² = [A / (x² + 1)] + [B / (x² + 1)²]

x³ + x² + 4x + 7 = A(x² + 1)² + B(x² + 1)

x³ + x² + 4x + 7 = A(x⁴ + 2x² + 1) + Bx² + B

x³ + x² + 4x + 7 = Ax⁴ + 2Ax² + A + Bx² + B

A = 0 (there is no x⁴), which means

A + B = 7 requires B = 7, which contradicts the quadratic term.

Proof your post for accuracy.