Question

convert to set  notation  x^3 +9x^2 - 108 ≥ 0

Answer 1

Rational factors of x³+9x²-108 are found by looking at the factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. Start with plugging in x=1: 1+9-108≠0; x=2: 8+36-108≠0; x=3: 27+81-108=0, so 3 is a zero and x-3 is a factor. Divide by this zero using synthetic division:

3 | 1  9   0  -108

     1  3 36 | 108

     1 12 36 | 0    = x²+12x+36=(x+6)².

Therefore (x+6)²(x-3)≥0 is another way of writing the inequality.

(x+6)² is always positive, so the whole expression is positive provided x-3≥0, that is, x≥3.

So the solution is x≥3, that is, { x≥3 }, the set of all numbers at least as big as 3.