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!

StarsIsTheLimit Feb 10, 2021

#4**0 **

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.

Guest Feb 10, 2021

#6**+1 **

Thank you for answering despite me not telling what degree it is. I appreciate it very much!

StarsIsTheLimit
Feb 10, 2021