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Given-f-x-x-5-ax-4-bx-3-cx-2-dx-c-and-f-1-f-2-f-3-f-4-f-5-Find-a-




Question Number 188651 by cortano12 last updated on 04/Mar/23
  Given f(x)=x^5 +ax^4 +bx^3 +cx^2 +dx+c   and f(1)=f(2)=f(3)=f(4)=f(5).   Find a.
$$\:\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{5}} +\mathrm{ax}^{\mathrm{4}} +\mathrm{bx}^{\mathrm{3}} +\mathrm{cx}^{\mathrm{2}} +\mathrm{dx}+\mathrm{c} \\ $$$$\:\mathrm{and}\:\mathrm{f}\left(\mathrm{1}\right)=\mathrm{f}\left(\mathrm{2}\right)=\mathrm{f}\left(\mathrm{3}\right)=\mathrm{f}\left(\mathrm{4}\right)=\mathrm{f}\left(\mathrm{5}\right). \\ $$$$\:\mathrm{Find}\:\mathrm{a}. \\ $$
Answered by horsebrand11 last updated on 04/Mar/23
 f(x)=(x−1)(x−2)(x−3)(x−4)(x−5)+p   f(x)=x^5 −15x^4 +85x^3 −225x^2 +274x−120+p  ⇒a=−15, b=85 , c=−225, d=274,   ⇒−225=−120+p  ⇒p=−105  ∴ f(x)=(x−1)(x−2)(x−3)(x−4)(x−5)−105
$$\:{f}\left({x}\right)=\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{5}\right)+{p} \\ $$$$\:{f}\left({x}\right)={x}^{\mathrm{5}} −\mathrm{15}{x}^{\mathrm{4}} +\mathrm{85}{x}^{\mathrm{3}} −\mathrm{225}{x}^{\mathrm{2}} +\mathrm{274}{x}−\mathrm{120}+{p} \\ $$$$\Rightarrow{a}=−\mathrm{15},\:{b}=\mathrm{85}\:,\:{c}=−\mathrm{225},\:{d}=\mathrm{274},\: \\ $$$$\Rightarrow−\mathrm{225}=−\mathrm{120}+{p} \\ $$$$\Rightarrow{p}=−\mathrm{105} \\ $$$$\therefore\:{f}\left({x}\right)=\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{5}\right)−\mathrm{105} \\ $$$$ \\ $$

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