find-such-a-polynomial-if-is-divided-by-x-2-then-the-remainder-is-5-if-it-isdivided-by-x-3-the-remainder-is-9-if-it-is-divided-by-x-4-the-remainder-is-13-if-divide-by-x-10-and-the-remaid

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Question Number 94818 by student work last updated on 21/May/20

$$\mathrm{find}\mathrm{such}\mathrm{a}\mathrm{polynomial}\mathrm{if}\mathrm{is}\mathrm{divided}\mathrm{by}(\mathrm{x}-2)\mathrm{then}\mathrm{the}$$$$\mathrm{remainder}\mathrm{is}5,\mathrm{if}\mathrm{it}\mathrm{isdivided}\mathrm{by}(\mathrm{x}-3)\mathrm{the}\mathrm{remainder}$$$$\mathrm{is}9,\mathrm{if}\mathrm{it}\mathrm{is}\mathrm{divided}\mathrm{by}(\mathrm{x}-4)\mathrm{the}\mathrm{remainder}\mathrm{is}13,\mathrm{if}$$$$\mathrm{divide}\mathrm{by}(\mathrm{x}-10)\mathrm{and}\mathrm{the}\mathrm{remaider}\mathrm{becomes}37\mathrm{and}$$$$\mathrm{if}(\mathrm{x}-\frac{3}{4})\mathrm{divided}\mathrm{by}\mathrm{the}\mathrm{remainder}\mathrm{becomes}\mathrm{zero}?$$

Commented by Rasheed.Sindhi last updated on 21/May/20

$$Therequiredpolynomialis4x-3$$

Commented by student work last updated on 21/May/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{practice}\mathrm{der}?$$

Commented by Rasheed.Sindhi last updated on 21/May/20

$$Ishallwriteanswerindetailsoon.$$

Commented by student work last updated on 21/May/20

$$\mathrm{you}\mathrm{show}\mathrm{practice}\mathrm{to}\mathrm{i}\mathrm{learn}$$

Commented by mr W last updated on 21/May/20

$$4x-3iscorrect,butnottheonlyone$$$$solution,ithink.$$$$i^{\prime }llpostmysolutionlater.$$

Commented by Rasheed.Sindhi last updated on 21/May/20

$$Yessir{it}^{\prime }slinearanswer,and$$$$if{I}^{\prime }vedonecorrectcalculation$$$$quadraticisnotpossible.Now{I}^{\prime }m$$$$tryingforcubicsolution\dots$$

Commented by student work last updated on 21/May/20

$$\mathrm{please}\mathrm{post}\mathrm{me}\mathrm{solution}\mathrm{sir}$$

Answered by mr W last updated on 21/May/20

$$saythepolynomialsearchedisf\left(x\right).$$$$f\left(x\right)canbedividedby(x-\frac{3}{4})without$$$$remainder,therefore$$$$f\left(x\right)=g\left(x\right)(x-\frac{3}{4})$$$$withg\left(x\right)beinganotherpolynomial.$$$$$$$$whenf\left(x\right)isdividedby(x-2)weget$$$$remainder5,thatmeans$$$$f\left(x\right)=p\left(x\right)(x-2)+5,i.e.f\left(2\right)=5,$$$$f\left(2\right)=g\left(2\right)(2-\frac{3}{4})=5\Rightarrow g\left(2\right)=4$$$$similarly$$$$f\left(3\right)=g\left(3\right)(3-\frac{3}{4})=9\Rightarrow g\left(3\right)=4$$$$f\left(4\right)=g\left(4\right)(4-\frac{3}{4})=13\Rightarrow g\left(4\right)=4$$$$f\left(10\right)=g\left(10\right)(10-\frac{3}{4})=37\Rightarrow g\left(10\right)=4$$$$thatmeans$$$$g\left(x\right)=k\left(x\right)(x-2)(x-3)(x-4)(x-10)+4$$$$withk\left(x\right)beinganypolynomial.$$$$\Rightarrow f\left(x\right)=g\left(x\right)(x-\frac{3}{4})$$$$\Rightarrow f\left(x\right)=[k\left(x\right)(x-2)(x-3)(x-4)(x-10)+4](x-\frac{3}{4})$$$$or$$$$\Rightarrow f\left(x\right)=[c\left(x\right)(x-2)(x-3)(x-4)(x-10)+1](4x-3)$$$$withc\left(x\right)beinganypolynomial$$$$\left(includinganyconstant\right)$$

Commented by student work last updated on 21/May/20

$$\mathrm{thanks}\mathrm{dear}$$

Commented by Rasheed.Sindhi last updated on 21/May/20

$$\mathcal{G}\mathit{r}\mathit{e}\mathit{a}\mathit{t}\mathit{m}\mathit{r}\mathcal{W}\mathit{s}\mathit{i}\mathit{r}!$$

Commented by Rasheed.Sindhi last updated on 21/May/20

$$f\left(x\right)iseitherlinearorithasdegree$$$$greaterthan4?$$

Commented by mr W last updated on 21/May/20

$$correctsir!$$$$correctsir!$$

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