Math, asked by HARSHA691, 5 hours ago

partial fraction
what are the steps for getting A B C values out their?​

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Answers

Answered by senboni123456
1

Step-by-step explanation:

We have,

 \rm \frac{ {x}^{2}  + 1}{ {x}^{3} + 3 {x}^{2}  + 3x + 2 }  =  \frac{ A }{x + 2}  +  \frac{B}{ {x}^{2} + x + 1 } +  \frac{C}{(x + 1)( {x}^{2} + x + 1) }   \\

 \rm \implies \frac{ {x}^{2}  + 1}{ {x}^{3} + 3 {x}^{2}  + 3x + 2 }  =   \frac{ A( {x}^{2} + x + 1) + B(x + 2) +  C}{(x + 2)( {x}^{2} + x + 1) }   \\

 \rm \implies \frac{ {x}^{2}  + 1}{ {x}^{3} + 3 {x}^{2}  + 3x + 2 }  =   \frac{ A {x}^{2} +Ax  + A + Bx + 2B +  C}{ {x}^{2} (x + 2) +   x (x + 2)+ 1(x + 2) }   \\

 \rm \implies \frac{ {x}^{2}  + 1}{ {x}^{3} + 3 {x}^{2}  + 3x + 2 }  =   \frac{ A {x}^{2} +Ax  + Bx  +A+ 2B +  C}{ {x}^{3} + 2 {x}^{2}  +    {x}^{2}  + 2x+ x + 2 }   \\

 \rm \implies \frac{ {x}^{2}  + 1}{ {x}^{3} + 3 {x}^{2}  + 3x + 2 }  =   \frac{ A {x}^{2} +(A  + B)x  +(A+ 2B +  C)}{ {x}^{3} + 3{x}^{2}   + 3x+ 2 }   \\

 \rm \implies {x}^{2}  + 1  =    A {x}^{2} +(A  + B)x  +(A+ 2B +  C)  \\

On comparing the cooeficients,

A = 1 \\ A + B = 0 \\ A + 2B + C = 1

 \implies 1 + B = 0  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:\\ \implies \ 1 + 2B + C = 1

 \implies  B =  - 1  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:\\ \implies \ 1 + 2B + C = 1

So,

1+ 2( - 1)+ C = 1

 \implies  - 2+ C = 0

 \implies   C = 2

So,

 \tt \purple{A - B + C = (1) - ( - 1) + 2}

 \tt  \purple{ \implies \: A - B + C = 1  + 1 + 2 = 4}

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