Sample quiz on factoring cubic and quartic polynomials
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1. Let $f(x)$ be a polynomial. If $f(a)=0$, which of the following is definitely true?
   
   $x+a$ is a factor of $f(x)$
   $x-a$ is a factor of $f(x)$
   $x-2a$ is a factor of $f(x)$
   $x+2a$ is a factor of $f(x)$
   
2. For what value of $k$ are both $x-1$ and $x+1$ factors of the cubic $x^3+kx$?
   
   $k=1$
   $k=2$
   $k=-2$
   $k=-1$
3. Is it always possible to factor every polynomial over the integers?
   
   Yes, by using the factor theorem
   Yes, by using the remainder theorem
   No, it is never possible
   No, it is sometimes impossible
4. In a bid to factor $x^3-2x^2+3x-12$, which of the following values of $x$ is NOT worth testing according to the integral root theorem?
   
   $8$
   $6$
   $4$
   $2$
5. Factorize $x^4-1$ completely
   
   $(x-1)^4$
   $(x+1)^2(x-1)^2$
   $(x-1)(x+1)(x^2+1)$
   $(x^2+1)(x-1)(x-1)$
6. Factorize $(x-1)^3-125$
   
   $(x-6)(x^2+3x-21)$
   $(x+6)(x^2+3x+21)$
   $(x-6)(x^2+3x+21)$
   $(x+6)(x^2-3x-21)$
7. Factorize $x^3-x^2-x+1$
   
   $(x-1)^{2}(x+1)$
   $(x+1)^{2}(x-1)$
   $(x-1)^3$
   $(x+1)^3$
8. Factorize $x^4-8x^2+16$
   
   $(x-4)^{2}(x+4)^{2}$
   $(x-2)^{2}(x+2)^{2}$
   $(x-4)^{2}(x+1)^{2}$
   $(x-1)^{2}(x+4)^{2}$
9. What is the value of $k$ for which $x+2$ is a factor of the quartic polynomial $x^4+2x^3-2x+k$?
   
   $k=-28$
   $k=-4$
   $k=28$
   $k=4$
10. Factorize $x^3-3x^2-4x+12$ completely
    
    $(x^2-4)(x-3)$
    $(x^2-4)(x+3)$
    $(x-2)(x+2)(x+3)$
    $(x+2)(x-2)(x-3)$