Answer to Question #325139 in Calculus for Busi

Question #325139

Decompose



(i)



(x^2+X+1)/(X+3)(x^2-x+1)



into partial fractions (show all the steps).


1
Expert's answer
2022-04-07T17:04:09-0400

x2+x+1(x+3)(x2x+1)=\frac{x^2+x+1}{(x+3)(x^2-x+1)}= Ax+3+Bx+Cx2x+1\frac{A}{x+3}+\frac{Bx+C}{x^2-x+1}

x2+x+1=A(x2x+1)+(Bx+C)(x+3)x^2+x+1=A(x^2-x+1)+(Bx+C)(x+3)

Equating coefficients, we get:

1=A+B1=A+B (1)

1=A+3B+C1=-A+3B+C (2)

1=A+3C1=A+3C (3)

Substituting (1) into (2) gives

1=A+3(1A)+C=4A+C+31=-A+3(1-A)+C=-4A+C+3 (4)

Now let’s substitute (3) into (4)

1=4(13C)+C+3=13C11=-4(1-3C)+C+3=13C-1

C=213C=\frac{2}{13}

A=13C=1613=713A=1-3C=1-\frac{6}{13}=\frac{7}{13}

B=1A=1713=613B=1-A=1-\frac{7}{13}=\frac{6}{13}

x2+x+1(x+3)(x2x+1)=\frac{x^2+x+1}{(x+3)(x^2-x+1)}=1137x+3+1136x+2x2x+1\frac{1}{13}\frac{7}{x+3}+\frac{1}{13}\frac{6x+2}{x^2-x+1}


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