Question 1209687: Find a polynomial f(x) of degree 5 such that both of these properties hold:
* f(x) is divisible by x^3.
* f(x) is divisible by (x+1)^3.
Write your answer in expanded form.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Since $f(x)$ is divisible by $x^3$, we can write $f(x) = x^3 g(x)$ for some polynomial $g(x)$.
Since $f(x)$ is divisible by $(x+1)^3$, we can write $f(x) = (x+1)^3 h(x)$ for some polynomial $h(x)$.
Since $f(x)$ is divisible by both $x^3$ and $(x+1)^3$, and these factors have no common roots, $f(x)$ must be divisible by their product, $x^3(x+1)^3$.
Since we are looking for a polynomial of degree 5, and the product $x^3(x+1)^3$ has degree 6, we must have a constant factor. Let's call it $a$. So, $f(x) = ax^3(x+1)^3$.
Since $f(x)$ is of degree 5, we must have $f(x) = ax^3(x+1)^2$ or $f(x) = a x^2(x+1)^3$. However, we are told that $f(x)$ is divisible by $x^3$ and $(x+1)^3$, so we must have $f(x) = ax^3(x+1)^3$. But this gives a polynomial of degree 6. To get degree 5, we can't have both $x^3$ and $(x+1)^3$ as factors.
The problem states that $f(x)$ *is* divisible by *both* $x^3$ and $(x+1)^3$. This implies that $f(x)$ must be divisible by $x^3(x+1)^3$, which is degree 6. The problem also states that $f(x)$ has degree 5. This is a contradiction. There is no such polynomial.
However, if the question meant to say that the degree is *at most* 5, then we can proceed as follows:
If $f(x)$ is divisible by $x^3$ and $(x+1)^3$, then $f(x)$ must be of the form $ax^3(x+1)^3$. However, this is a degree 6 polynomial.
If the degree of $f(x)$ is *at most* 5, and $f(x)$ is divisible by $x^3$ and $(x+1)^2$, then $f(x) = ax^3(x+1)^2$.
If the degree of $f(x)$ is *at most* 5, and $f(x)$ is divisible by $x^2$ and $(x+1)^3$, then $f(x) = ax^2(x+1)^3$.
If the degree of $f(x)$ is *at most* 5, and $f(x)$ is divisible by $x^3$ and $(x+1)$, then $f(x) = ax^3(x+1)^2$.
If the degree of $f(x)$ is *at most* 5, and $f(x)$ is divisible by $x$ and $(x+1)^3$, then $f(x) = ax(x+1)^3$.
Assuming that the degree is exactly 5 and f(x) is divisible by both $x^3$ and $(x+1)^3$ is impossible.
If the degree is at most 5, and f(x) is divisible by $x^3$ and $(x+1)^2$, then $f(x) = ax^3(x+1)^2 = ax^3(x^2+2x+1) = ax^5+2ax^4+ax^3$.
If the degree is at most 5, and f(x) is divisible by $x^2$ and $(x+1)^3$, then $f(x) = ax^2(x+1)^3 = ax^2(x^3+3x^2+3x+1) = ax^5+3ax^4+3ax^3+ax^2$.
Final Answer: The final answer is $\boxed{no such polynomial}$
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