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If g(x)-(f(x))/((x-a)(x-b)(x-c)),w h e r...

If `g(x)-(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x)` is a polynomial of degree `<3,` then prove that `(dg(x))/(dx)=|1af(a)(x-a)^(-2)1bf(b)(x-b)^(-2)1cf(c)(x-c)^(-2)|+|a^2a1b^2b1c^2c1|`

Text Solution

Verified by Experts

By partial fractions we have
`g(x)=(f(a))/((x-a)(a-b)(a-c))+(f(b))/((b-a)(x-b)(b-c))`
`+(f(c ))/((c-a)(c-b)(x-c))`
`=(1)/((a-b)(b-c)(c-a))xx`
`[(f(a)(c-b))/((x-a))+(f(b)(a-c))/((x-b))+(f(c )(b-a))/((x-c))]`
`= |{:(1,,a,,f(a)//(x-a)),(1,,b,,f(b)//(x-b)),(1,,c,,f(c )//(x-c)):}|-:|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|`
`rArr (dg(x))/(dx)= |{:(1,,a,,-f(a)(x-a)^(-2)),(1,,b,,-f(b)(x-b)^(-2)),(1,,c,,-f(c )(x-c)^(-2)):}|-:|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|`
`= |{:(1,,a,,f(a)(x-a)^(-2)),(1,,b,,f(b)(x-b)^(-2)),(1,,c,,f(c )(x-c)^(2)):}|-:|{:(a^(2),,a,,1),(b^(2),,b,,1),(c^(2),,c,,1):}|`
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