Question

2. Factor f(x) = x4 + x3 – 8x2 + 6x + 36 completely.

Answer 1

Take the leading coefficient and the last coefficient, and list all their factors. In this case, the leading coefficient is 1 because it is the number in front of the highest order term. 36 is the last coefficient.

$a_n$ = 1: 1
$a_0$ = 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 

Next we will find all the possible rational roots. To do so, divide each of the $a_0$ factors by each of the $a_n$ factors. For each result there is a positive and negative root. In this case 1 is the only $a_n$ factor. Since anything divided by 1 is itself, the possible rational roots are the same as the factors of 36 (except now we have positive and negative numbers).

So the possible roots are -1, 1, -2, 2, -3, 3, -4, 4, -6, 6.... etc.

Now use synthetic division to test each of the possible roots. If the answer does not have a remainder, then it is a root. After one possible root works, use the result of the division to continue finding other roots. Let me know if you need help with this step and I will upload a picture. Otherwise, division is pretty straight forward.

After dividing the polynomial by each possible rational root, we find -2 and -3 as roots. We can express these roots as the factors (x+2) and (x+3). So after wee factor them out, we get:

(x+2)(x+3)(x^2 - 4x + 6)

Hope this helps! Rational Root Theorem is a lot of work, but it is easy once you understand it!