find the rational zeros in x^3+6x+20

Question

I have no idea what im doing

Answer 1

x³+6x+20.

Factors of 20 1, 2, 4, 5, 10, 20. Since we only have plus signs in the expression we need only look at the negatives of these.

If we start with -1, then we would have -1-6+20=13, so x=-1, i.e., x+1 is not a factor.

Next, x=-2: -8-12+20=0, so x+2 is a factor. We can divide by -2 using synthetic division:

-2 | 1  0  6     20

      1 -2  4  | -20

      1 -2 10 |  0 = x²-2x+10 doesn't factorise to produce real factors (only complex).

(x+2)(x²-2x+10)=x³-2x²+10x+2x²-4x+20=x³+6x+20.

Additional factors are found by solving x²-2x+10=0:

x²-2x+1+9=0, x²-2x+1=-9, (x-1)²=-9, x-1=±√-9=±3i, and x=1+3i or 1-3i.