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Question Number 19811 by Joel577 last updated on 16/Aug/17

$$\mathrm{If}f\left(x\right)=(x+1)g\left(x\right)-2\mathrm{and}g\left(3\right)=4$$$$\mathrm{Find}\mathrm{the}\mathrm{remainder}\mathrm{if}f\left(x\right)\mathrm{divided}\mathrm{by}$$$$(x+1)(x-3)$$

Commented by myintkhaing last updated on 16/Aug/17

$$\mathrm{Since},\mathrm{f}\left(\mathrm{x}\right)=(\mathrm{x}+1)\mathrm{g}\left(\mathrm{x}\right)-2\mathrm{and}\mathrm{g}\left(3\right)=4$$$$\mathrm{when}\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}\mathrm{divided}\mathrm{by}(\mathrm{x}+1),$$$$\mathrm{the}\mathrm{remainder},\mathrm{f}(-1)=-2\mathrm{and}$$$$\mathrm{f}\left(3\right)=4\mathrm{g}\left(3\right)-2=14$$$$\mathrm{Let}\mathrm{the}\mathrm{quotient}\mathrm{be}\mathrm{Q}\left(\mathrm{x}\right)\mathrm{and}\mathrm{the}\mathrm{remainder}$$$$\mathrm{be}\mathrm{ax}+\mathrm{b}$$$$\mathrm{when}\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}\mathrm{divided}\mathrm{by}(\mathrm{x}+1)(\mathrm{x}-3)$$$$\mathrm{So},\mathrm{f}\left(\mathrm{x}\right)=(\mathrm{x}+1)(\mathrm{x}-3)\mathrm{Q}\left(\mathrm{x}\right)+\mathrm{ax}+\mathrm{b}$$$$\mathrm{thus}\mathrm{f}(-1)=-\mathrm{a}+\mathrm{b}=-2.........\left(1\right)$$$$\mathrm{f}\left(3\right)=3\mathrm{a}+\mathrm{b}=14.........\left(2\right)$$$$\mathrm{Solving}\mathrm{equations}\left(1\right)\mathrm{and}\left(2\right)$$$$\mathrm{a}=4\mathrm{and}\mathrm{b}=2$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

Commented by Joel577 last updated on 16/Aug/17

$$thankyouverymuch$$

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