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Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold:  $\bullet$ $f(x)$ is divisible by $x^3$.  $\bullet$ $f(x)+2$ is divisible by $(x+1)^3$.

 Jul 26, 2020
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Since f(x) is divisible by x^3, f(x) is of the form ax^5 + bx^4 + cx^3.

 

You then want ax^5 + bx^4 + cx^3 + 2 to be divisible by (x + 1)^3.  Using long division, you get the equations

-10a  + 6b - 3c = 0

4a - 3b + 2c = 0

-a + b - c + 2 = 0

==> a = 6, b = 16, c = 12

 

So f(x) = 6x^5 + 16x^4 + 12x^3.

 Jul 26, 2020

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