Prove that (ax^(2))/((x -a)(x-b)(x-c))...

Prove that
`(ax^(2))/((x -a)(x-b)(x-c))+(bx)/((x -b)(x-c))+(c)/(x-c)+1`
`= (x^(3))/((x-a)(x-b)(x-c))`.

Text Solution

Verified by Experts

`(ax^(2))/((x -a)(x-b)(x-c))+(bx)/((x -b)(x-c))+(c)/(x-c)+1`
`=(ax^(2))/((x-a)(x-b)(x-c))+(bx)/((x -b)(x-c))+((c +x-c)/(x-c))`
`=(ax^(2))/((x-a)(x-b)(x-c))+(bx)/((x -b)(x-c))+(x)/(x-c)`
`=(ax^(2))/((x-a)(x-b)(x-c))+(bx +x(x-b))/((x-b)(x-c))`
`=(ax^(2))/((x-a)(x-b)(x-c))+(x^(2))/((x-b)(x-c))`
`=(ax^(2)+x^(2)(x-a))/((x-a)(x-b)(x-c))`
`=(x^(3))/((x-a)(x-b)(x-c))`

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