Problem 2

Divide 36*a2+1-64*a4-12*a by 6*a-1-8*a2.

Solution
a) Rewrite in standard form. The numerator becomes

-64*a4+36*a2-12*a+1

and the denominator becomes

-8*a2+6*a-1.

b) Factor the denominator by pulling out a negative sign:

-8*a2+6*a-1=-(8*a2-6*a+1)

Thus, the division is equivalent to

(-64*a4+36*a2-12*a+1)/(-(8*a2-6*a+1))=(64*a4-36*a2+12*a-1)/(8*a2-6*a+1)

c)Set up the division. We expect a quadratic quotient Q(a)=C*a2+D*a+E such that

(8*a2-6*a+1)*(C*a2+D*a+E)=64*a4-36*a2+12*a-1.

d) Multiply out and equate coefficients. The product expands to

8*C*a4+(8*D-6*C)*a3+(8*E-6*D+C)*a2+(-6*E+D)*a+E

Equate this with 64*a4+0*a3-36*a2+12*a-1.

For a4: 8*C=64 so C=8.

For a3: 8*D-6*C=0 yields 8*D-48=0 so D=6.

For a2: 8*E-6*D+C=-36 gives 8*E-36+8=-36 so 8*E=-8 and E=-1.

For the a term: -6*E+D=12 with E=-1 and D=6 this checks.

The constant term gives E=-1 which matches.

Answer: 8*a2+6*a-1