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Find a polynomial \(f(x)\) of degree  such that both of these properties hold:

\(f(x)\) is divisible by \(x^3\).

 

\(f(x) + 2\) is divisible by \((x+1)^3\).

 

Thank you!

 Feb 10, 2021
 #1
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of degree what?  Is there supposed to be a number there?

 Feb 10, 2021
 #3
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It's a repeat question from about a week ago.

Guest Feb 10, 2021
 #4
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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.

 Feb 10, 2021
 #5
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That is nicely thought through guest.   laugh

 

(I haven't checked the final answer)

Melody  Feb 10, 2021
 #6
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Thank you for answering despite me not telling what degree it is. I appreciate it very much! laugh

StarsIsTheLimit  Feb 10, 2021

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