2x^(5) + 3x^(4) – 30x^(3) – 57x^(2) – 2x + 24
is there a possibility that we have to factor something like this on the test, if so, how do you do it?? it just looks scary
First find P/Q where P = factors of 24 and Q = factors of 2. Remember that P/Q can be + or -.
2x^(5) + 3x^(4) – 30x^(3) – 57x^(2) – 2x + 24
You have P = +- (1, 2, 3, 4, 6, 8, 12, 24) and Q = +- (1, 2). Your choices are as follows:
+- (1, 2, 3, 4, 6, 8, 12, 24, 1/2, 3/2)
Unfortunately you'll have to go down the list and furthermore there may not be a rational number as a root. Just pray.
Synthetic division works wonders. Look at that link.
I tried 4 first and got lucky since it works. I ended up with 2, 11, 14, -1, and -6. This translates into 2x^(4) + 11x^(3) + 14x^(2) - x - 6. Still looks ugly.
At this point you can narrow down the list.
P = +- (1, 2, 3, 6) and Q = +- (1, 2)
P/Q = +- (1, 2, 3, 6, 1/2, 3/2)
The next one that worked was -1. I ended up with 2, 9, 5, and -6. This translates into 2x^(3) + 9x^(2) + 5x - 6. Looks nicer but still pretty bad.
List stays the same because P and Q are the same. I got -3/2 to work next. I ended up with 2, 6, and -4 which is the same thing as 1, 3, and -2. This translates to x^2 + 3x - 2.
From here we're stuck with the quadratic formula. Turns out you get -3 +- sqrt(17) all over 2. Those are your last 2 roots.
They don't show up with P/Q because they are irrational numbers (square roots of non perfect squares).
