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Question Number 188651 by cortano12 last updated on 04/Mar/23

$$\mathrm{Given}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^5+{\mathrm{ax}}^4+{\mathrm{bx}}^3+{\mathrm{cx}}^2+\mathrm{dx}+\mathrm{c}$$$$\mathrm{and}\mathrm{f}\left(1\right)=\mathrm{f}\left(2\right)=\mathrm{f}\left(3\right)=\mathrm{f}\left(4\right)=\mathrm{f}\left(5\right).$$$$\mathrm{Find}\mathrm{a}.$$

Answered by horsebrand11 last updated on 04/Mar/23

$$f\left(x\right)=(x-1)(x-2)(x-3)(x-4)(x-5)+p$$$$f\left(x\right)=x^5-15x^4+85x^3-225x^2+274x-120+p$$$$\Rightarrow a=-15,b=85,c=-225,d=274,$$$$\Rightarrow -225=-120+p$$$$\Rightarrow p=-105$$$$\therefore f\left(x\right)=(x-1)(x-2)(x-3)(x-4)(x-5)-105$$$$$$

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