Express ` (7x+3)/(x-1)` in the form `A + B/(x-1)` , where A and B are constants.
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To express the given rational expression in the form A + B/(x-1), divide the numerator by the denominator using long division.
` x - 1 | bar(7x+3) `
The steps are:
> Divide 7x by x. `(7x)/x=7`
Write the 7 at the top of 7x.
> Then, multiply the resulting value by x-1. `7(x-1)=7x-7`
Write this below 7x + 3.
> Next, subtract this from 7x+3. `7x+3-(7x-7)=11`
Since 11 has no x, this is the remainder.
> And, express the remainder as a fraction and add it to 7.
`x - 1 | bar(7x+3) `
`(-)` `7x - 7`
Hence, `(7x+3)/(x-1) = 7 + 11/(x-1)` .
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