Question

Resolve x^3/(x-a)(x-b)(x-c) into partial fraction​

Answer 1

Answer:

\bf{\frac{x}{(x-a)(x-b)(x-c)}=\frac{a}{(a-b)(a-c)}\frac{1}{x-a}+\frac{b}{(b-a)(b-c)}\frac{1}{x-b}+\frac{c}{(c-a)(c-b)}\frac{1}{x-c}}

Step-by-step explanation:

Resolve x/(x-a)(x-b)(x-c) into partial fraction

\text{let}\frac{x}{(x-a)(x-b)(x-c)}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}

\implies\:\frac{x}{(x-a)(x-b)(x-c)}=\frac{A(x-b)(x-c)+B(x-a)(x-c)+C(x-a)(x-b)}{(x-a)(x-b)(x-c)}

\implies\:x=A(x-b)(x-c)+B(x-a)(x-c)+C(x-a)(x-b)..............(1)

put x=a in (1), we get

\implies\:a=A(a-b)(a-c)

\implies\:\bf{A=\frac{a}{(a-b)(a-c)}}

put x=b in (1), we get

\implies\:b=B(b-a)(b-c)

\implies\:\bf{B=\frac{b}{(b-a)(b-c)}}

put x=c in (1), we get

\implies\:c=C(c-a)(c-b)

\implies\:\bf{C=\frac{c}{(c-a)(c-b)}}

\therefore\:\bf{\frac{x}{(x-a)(x-b)(x-c)}=\frac{a}{(a-b)(a-c)}\frac{1}{x-a}+\frac{b}{(b-a)(b-c)}\frac{1}{x-b}+\frac{c}{(c-a)(c-b)}\frac{1}{x-c}}

Answer 2

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