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Reil [10]
3 years ago
14

Resolve into partial fractions​

Mathematics
1 answer:
lisabon 2012 [21]3 years ago
3 0

Answer:

See below

Step-by-step explanation:

1)

\frac{11 + x}{(2 - x)(x - 3)}  =  \frac{A}{2 - x}  +  \frac{B}{x - 3}  \\  \\  \therefore \:  \frac{11 + x}{(2 - x)(x - 3)}  =  \frac{A(x - 3) +B(2 - x) }{(2 - x)(x - 3)}   \\  \\ \therefore \:  11 + x = Ax - 3A + 2B - Bx \\  \\  \therefore \: 11 + x = - 3A  + 2B  + Ax -  Bx\\  \\  \therefore \: 11 + x = - 3A  + 2B  + (A - B)x \\  \\ equating \: the \: like \: terms \: on \: both \: sides \\ A - B = 1 \\  \therefore \: A  =  B  +  1.....(1) \\  \\  -  3A  + 2B = 11....(2) \\  from \: eq \: (1) \: and \: (2) \\  - 3(B  +  1) + + 2B = 11 \\  \\ - 3B   - 3  + 2B = 11 \\  \\  - B = 11 + 3 \\  \\ B =  - 14  \\ A  =   - 14 +  1 =  - 13 \\  \\  \frac{11 + x}{(2 - x)(x - 3)}  =  \frac{ - 13}{2 - x}  +  \frac{ - 14}{x - 3} \\  \\ \frac{11 + x}{(2 - x)(x - 3)}  = -   \frac{ 13}{2 - x}   -  \frac{ 14}{x - 3}

2)

\frac{12x + 11}{x^2 +x - 6}

=\frac{12x + 11}{x^2 +3x-2x - 6}

=\frac{12x + 11}{x(x +3) -2(x +3)}

\therefore \frac{12x + 11}{x^2 +x - 6}=\frac{12x + 11}{(x +3) (x - 2)}

\therefore \frac{12x + 11}{x^2 +x - 6}=\frac{A}{(x +3)}+\frac{B}{(x - 2)}

\therefore \frac{12x + 11}{x^2 +x - 6}=\frac{A(x-2)+B(x+3)}{(x-2)(x+3)}

\therefore 12x + 11 = A(x-2)+B(x+3)

\therefore 12x + 11 = Ax-2A+Bx+3B

\therefore 12x + 11 = Ax+Bx-2A+3B

\therefore 12x + 11 = (A+B) x-2A+3B

Equating like terms on both sides:

A + B = 12\implies A = 12-B... (1)

- 2A + 3B = 11... (2)

Solving equations (1) & (2), we find:

A = 5, B = 7

\therefore \frac{12x + 11}{x^2 +x - 6}=\frac{5}{(x +3)}+\frac{7}{(x - 2)}

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