We are asked to factor
x^7+c^3x^4-c^4x^3-c^7
Not so easy this one.
First note that if x=c then the expression yields zero.
c^7+c^3c^4-c^4c^3-c^7=c^7+c^7-c^7-c^7=0
This means that (x-c) is a factor of x^7+c^3x^4-c^4x^3-c^7.
Let's do the long division and see what remains.
x^7+c^3x^4-c^4x^3-c^7
=x^7-cx^6+cx^6-c^2x^5+c^2x^5-c^3x^4+2c^3x^4-2c^4x^3+c^4x^3-c^5x^2+c^5x^2-c^6x+c^6x-c^7
=x^6(x-c)+cx^5(x-c)+c^2x^4(x-c)+2c^3x^3(x-c)+c^4x^2(x-c)+c^5x(x-c)+c^6(x-c)
=(x^6+cx^5+c^2x^4+2c^3x^3+c^4x^2+c^5x+c^6)(x-c)
Now we notice that when x=-c,
x^6+cx^5+c^2x^4+2c^3x^3+c^4x^2+c^5x+c^6=c^6-c^6+c^6-2c^6+c^6-c^6+c^6=0
Let's divide x^6+cx^5+c^2x^4+2c^3x^3+c^4x^2+c^5x+c^6 by (x+c).
x^6+cx^5+c^2x^4+2c^3x^3+c^4x^2+c^5x+c^6
=x^6+cx^5+c^2x^4+c^3x^3+c^3x^3+c^4x^2+c^5x+c^6
=x^5(x+c)+c^2x^3(x+c)+c^3x^2(x+c)+c^5(x+c)
=(x^5+c^2x^3+c^3x^2+c^5)(x+c)
Next notice that when x=-c
x^5+c^2x^3+c^3x^2+c^5=-c^5-c^5+c^5+c^5=0
So lets divide x^5+c^2x^3+c^3x^2+c^5 by (x+c).
x^5+c^2x^3+c^3x^2+c^5
=x^5+cx^4-cx^4-c^2x^3+2c^2x^3+2c^3x^2-c^3x^2-c^4x+c^4x+c^5
=x^4(x+c)-cx^3(x+c)+2c^2x^2(x+c)-c^3x(x+c)+c^4(x+c)
=(x^4-cx^3+2c^2x^2-c^3x+c^4)(x+c)
Now we are forced to factor x^4-cx^3+2c^2x^2-c^3x+c^4.
A little "inspired grouping" gets us over this hurdle.
Start with
x^4-cx^3+2c^2x^2-c^3x+c^4.
Split the third term.
x^4-cx^3+c^2x^2+c^2x^2-c^3x+c^4
Commute the 3rd term.
x^4+c^2x^2-cx^3-c^3x+c^2x^2+c^4
Group and factor.
x^2(x^2+c^2)-cx(x^2+c^2)+c^2(x^2+c^2)
(x^2-cx+c^2)(x^2+c^2)
If c and x are both real numbers, then we cannot factor any further. Note that the only way for x^2+c^2=0 is if they are both zero or if either x^2 or c^2 are less than zero. The only way for to have a squared number be less than zero is if it is imaginary. Also note that by using the quadratic formula, we can find x's that satisfy x^2-cx+c^2=0.
x=(cpmsqrt(c^2-4c^2))/2=(cpmcsqrt(-3))/2
So both roots for this expression MUST be imaginary regardless of whether c is imaginary or not.
So we have the factored form with the following logic:
x^7+c^3x^4-c^4x^3-c^7
=(x^6+cx^5+c^2x^4+2c^3x^3+c^4x^2+c^5x+c^6)(x-c)
=(x^5+c^2x^3+c^3x^2+c^5)(x+c)(x-c)
=(x^4-cx^3+2c^2x^2-c^3x+c^4)(x+c)(x+c)(x-c)
=(x^2-cx+c^2)(x^2+c^2)(x+c)(x+c)(x-c)
=(x^2-cx+c^2)(x^2+c^2)(x+c)^2(x-c)
Yikes! Your teacher sure is mean!