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Question Number 48208 by naka3546 last updated on 20/Nov/18

$$f\left(x\right)+{(x+1)}^3=2f(x+1)$$$$f\left(10\right)=?$$

Answered by MJS last updated on 20/Nov/18

$$f\left(x\right)={ax}^3+{bx}^2+cx+d$$$$f\left(x\right)+{(x+1)}^3=$$$$=(a+1)x^3+(b+3)x^2+(c+3)x+(d+1)$$$$2f(x+1)=$$$$=2{ax}^3+2(3a+b)x^2+2(3a+2b+c)x+2(a+b+c+d)$$$$\mathrm{comparing}\mathrm{the}\mathrm{constants}$$$$a+1=2a$$$$b+3=2(3a+b)$$$$c+3=2(3a+2b+c)$$$$d+1=2(a+b+c+d)$$$$\Rightarrow a=1,b=-3,c=9,d=-13$$$$f\left(x\right)=x^3-3x^2+9x-13$$$$f\left(10\right)=777$$

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