3x^2+17x+14÷x^3-8 resolve into partial fraction
Answers
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Step-by-step explanation:
Factor the denominator: 3x2+17x+14x3−8=3x2+17x+14(x−2)(x2+2x+4)
The form of the partial fraction decomposition is
3x2+17x+14(x−2)(x2+2x+4)=Ax−2+Bx+Cx2+2x+4
Write the right-hand side as a single fraction:
3x2+17x+14(x−2)(x2+2x+4)=(x−2)(Bx+C)+(x2+2x+4)A(x−2)(x2+2x+4)
The denominators are equal, so we require the equality of the numerators:
3x2+17x+14=(x−2)(Bx+C)+(x2+2x+4)A
Expand the right-hand side:
3x2+17x+14=x2A+x2B+2xA−2xB+xC+4A−2C
Collect up the like terms:
3x2+17x+14=x2(A+B)+x(2A−2B+C)+4A−2C
The coefficients near the like terms should be equal, so the following system is obtained:
⎧⎩⎨A+B=32A−2B+C=174A−2C=14
Solving it (for steps, see system of equations calculator), we get that A=5, B=−2, C=3
Therefore,
3x2+17x+14(x−2)(x2+2x+4)=5x−2+3−2xx2+2x+4
Answer: 3x2+17x+14x3−8=5x−2+3−2xx2+2x+4