Sample quiz on division of polynomials and the remainder theorem
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1. When the polynomial $x^3-1$ is divided by $x-1$, what is the quotient?
   
   $x^2-x+1$
   $x^2+x+1$
   $x^2-2x+1$
   $x^2-2x-1$
   
2. What is the quotient when $2x^2-x-15$ is divided by $5x+11$?
   
   $\frac{2x}{5}-\frac{27}{25}$
   $\frac{2x}{5}+\frac{27}{25}$
   $\frac{2x}{5}-\frac{27}{5}$
   $\frac{2x}{5}+\frac{27}{5}$
3. Division of $3x^2-5x-2$ by $2x^3-x^2-x+1$ is not permissible because $\cdots\cdots$
   
   the divisor contains more terms than the dividend
   the divisor has a higher degree than the dividend
   the divisor is not factorable
   the dividend is not factorable
4. Is it possible for a polynomial to be divided by two different linear functions with the resulting remainders being equal?
   
   Yes, sometimes
   Yes, always the case
   No, never possible
   Not sure
5. The remainder theorem states that if a polynomial $f(x)$ is divided by $ax+b$, then the remainder is $\cdots\cdots$?
   
   $f(a/b)$
   $f(-b/a)$
   $f(-a/b)$
   $f(a+b)$
6. The synthetic method of dividing polynomials has one limitation, namely:
   
   it doesn't work for linear divisors
   it only works for linear divisors
   it is difficult to implement in practice
   it is not possible to write an algorithm for it
7. What is the remainder when $x^4-4x^3+12x^2-24x+24$ is divided by $x-1$?
   
   $-65$
   $-9$
   $65$
   $9$
8. After dividing the cubic $x^3+kx^2-4x+2$ by $x+2$, a remainder of $26$ was obtained. The value of $k$ is $\cdots\cdots$?
   
   $2$
   $6$
   $-6$
   $-2$
9. If the division of $2x^3-3x^2+kx-1$ by $x-1$ yields a remainder of $2$, what is the value of $k$?
   
   $-4$
   $-8$
   $8$
   $4$
10. Write $\frac{x^3-3x^2+6x-6}{x+2}$ in the form Polynomial = Divisor $~\times ~$ Quotient + Remainder
    
    $x^3-3x^2+6x-6=(x+2)(x^2+5x+16)+38$
    $x^3-3x^2+6x-6=(x+2)(x^2+5x+16)-38$
    $x^3-3x^2+6x-6=(x+2)(x^2-5x+16)-38$
    $x^3-3x^2+6x-6=(x+2)(x^2-5x-16)-38$