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1) Find a polynomial $q(x)$ such that $(x+1)^3+x^2\cdot q(x)$ has degree less than $2$.

2) Find all integers $n$ for which $\frac{n^2+n+1}{n-1}$ is an integer.

3) Find the quotient and remainder when $x^2+2$ is divided by $x-2$.

4) Find the quotient and remainder when $9x^4+x^3-12x+21$ is divided by $x+4$

5) Find the quotient and remainder when $t^4-t^2+3t-7$ is divided by $t^2-3t+8$.

6) The polynomial $4x^3 - 20x^2 + 19x + 15$ has a root at $x = 3$. Find the other two roots.

7) Given that $x=2$ is a root of $p(x)=x^4-3x^3+6x^2-12x+8$, find the other roots (real and nonreal).

8) When $x^3 + 30x - c$ is divided by $x - 5$ the remainder is 5. What is $c$?

9) When $x^4 + 6x^3 - ax^2 - 45 x - 15$ is divided by $x^2 - x - 6$ the remainder is 3. What is $a$?

10) Find the remainder when $r^{13} + 1$ is divided by $r - 1$.

 Apr 26, 2019

Seriously???    TEN questions in a single post?   No attempt on your part to answer ANY of these?   We are not here to give answers to your homework or test questions!   

  TRY to answer them ....THEN post a single question at a time....cheeky

 Apr 26, 2019

Have to agree with EP......even worse is that you already signed off.....


cool cool cool

 Apr 26, 2019

Person, please stop using this as a cheat site. It is already obvious at this point, you clearly have an inability to do your own homework so please stop using this as a cheat site and try solving these problems on your own please.



If you still need help you can always ask us to verify your answers or tell your what you have done wrong.

 Apr 26, 2019

Another AOPS student not following directions. Student, we will soon notify your parents of the misbehavior you are conducting, please refrain from seeking for others for answer. Thank you and please continue to listen to instruction.

 Apr 29, 2019

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