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Question Number 19101 by Tinkutara last updated on 04/Aug/17

$$\mathrm{A}\mathrm{polynomial}f\left(x\right)\mathrm{with}\mathrm{rational}$$$$\mathrm{coefficients}\mathrm{leaves}\mathrm{remainder}15,\mathrm{when}$$$$\mathrm{divided}\mathrm{by}x-3\mathrm{and}\mathrm{remainder}2x+1,$$$$\mathrm{when}\mathrm{divided}\mathrm{by}{(x-1)}^2.\mathrm{Find}\mathrm{the}$$$$\mathrm{remainder}\mathrm{when}f\left(x\right)\mathrm{is}\mathrm{divided}\mathrm{by}$$$$(x-3){(x-1)}^2.$$

Commented by prakash jain last updated on 04/Aug/17

$$f\left(x\right)=G\left(x\right)(x-3){(x-1)}^2+{ax}^2+bx+c$$$$f\left(3\right)=15$$$$15=9a+3b+c$$$$3=a+b+c$$$$8a+2b=12\Rightarrow 4a+b=6$$$${ax}^2+bx+c=k(x^2-2x+1)+2x+1$$$$k=a$$$$2-2a=b$$$$a=2,b=-2,c=3$$$$P\left(x\right)=G\left(x\right)(x-3){(x-1)}^2+2x^2-2x+3$$$$P\left(3\right)=15$$$$P\left(x\right)=G\left(x\right)(x-3){(x-1)}^2+2{(x-1)}^2+2x+1$$$$remainderwhenP\left(x\right)\mathrm{is}\mathrm{divided}\mathrm{by}$$$$(x-3){(x-1)}^2=2x^2-2x+3$$$$rationalcoefficientinformation$$$$isnotnecessary.$$

Commented by RasheedSindhi last updated on 05/Aug/17

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{Q}\left(\mathrm{x}\right){(\mathrm{x}-1)}^2+2\mathrm{x}+1$$$$\mathrm{f}\left(1\right)=2\left(1\right)+1=3$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{G}\left(\mathrm{x}\right)(\mathrm{x}-3){(\mathrm{x}-1)}^2+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$$$$\mathrm{f}\left(1\right)=\mathrm{a}{\left(1\right)}^2+\mathrm{b}\left(1\right)+\mathrm{c}$$$$3=\mathrm{a}+\mathrm{b}+\mathrm{c}$$

Commented by Tinkutara last updated on 05/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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