Wow.
Since the coefficient of x^4 is 1, we need only worry about factors of -24 in searching for possible roots of the given polynomial.
I'd use synthetic div. here:
4 / 1 -5 -52 -70 -24
4 -4 224 616
1 -1 -56 154 592
There is a non-zero remainder (592) which is a long way frorm zero, so 4 is NOT a root. Try synth. div. using other factors of -24: {1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24} until your syth. div. produces a zero remainder.
What I did next was to graph X^4-5x^3-52x^2-70x-24, I immediately found that -4 is a zero (root), so I did the synth. div. again using -4 as divisor:
-4 / 1 -5 -52 -70 -24
-4 36 64 24
1 -9 -16 -6 0
Thus, -4 has been confirmed to be a root of the given polynomial.
Next, look at the 4 coefficients in front of that 0 remainder. What are factors of -6? {1, -1, 2, -2, 3, -3, 6, -6)
Is -2 a root? Let's try it:
-2 / 1 -5 -52 -70 -24
-2 14 76 -12
1 -7 -38 6 -36 Here we have a remainder of -36, so we
conclude that -2 is not a root of the original
polynomial.
Try again. Let's see whether 3 is a root. I'll use the 4 coefficients we found earlier: 1 -9 -16 -6
3 / 1 -9 -16 -6
3 -18 -102
1 -6 -34 -108 No, 3 is not a root because the remainder is not zero.
Continue in this manner until you have identified the three roots of the polynomial whose coefficeints are 1 -9 -16 -6.
