Domain and solve

Question

(x-5)^2(x+1)^3(x-1) is greater than or equal to 0

i need to find domain and solution

Answer 1

Yefim's answer using trying points in each interval works.

Another possibility.

The leading term is 1x^6, so we know in the far left and far right interval, the function is positive. When a root has even multiplictity, the sign of the function is the same on both sides. If the root has odd multiplictity, the sign switches.

Thus, moving from right to left, the function is positive on (5, ∞). As 5 has even multiplicity (2), the function is positive on (1, 5). As 1 has odd multiplicity(1), the sign switches on (-1, 1), so it is negative. As -1 has odd multiplicity (3), the sign switches on (-∞, -1), so it is positive there (we already knew it was positive there because of the degree of the polynomial being even and this being the far left interval. Then, we include the roots, as we are looking for where the function is greater than or equal to 0.

[1, ∞) U [-∞, -1]

Answer 2

(x - 5)²(x + 1)³(x - 1) ≥ 0.

Boundary values x = 5, x = - 1 and x = 1.

We have 4 intervals: (- ∞, - 1], [- 1, 1], [1, 5], [5, ∞).

Test Points: -2, (- 2 - 5)²(- 2 + 1)³(- 2 - 1) = 147 > 0.

Now we have solution: (- ∞, - 1]U[1,∞)