Answer to Question #342459 in Calculus for Dkay

Question #342459

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


1
Expert's answer
2022-05-20T04:34:07-0400

"\\frac{x^2+x+1}{(x+3)(x^2-x+1)}=\\frac{A}{x+3}+\\frac{Bx+C}{x^2-x+1}"

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

Let's find A, B and C by equating corresponding coefficients:

"1=A+B" (1)

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

"1=A+3C" (3)

Substracting (3) from (1) we obtain:

"B=3C" (4)

Substitution (4) into (2) gives

"1=-A+10C" (5)

Adding (3) and (5) we obtain

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

"B=\\frac{6}{13}"

"A=\\frac{7}{13}"

Answer:

"\\frac{x^2+x+1}{(x+3)(x^2-x+1)}=\\frac{7}{13}\\frac{1}{x+3}+\\frac{2}{13}\\frac{3x+1}{x^2-x+1}"



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