5x^5-2x^4-25x^3+10x^2+20x-8<=0

Предыдущий
вопрос Следующий
вопрос

To solve this inequality, we need to find the roots of the polynomial and then determine the sign of the polynomial in each interval between the roots.

The given polynomial is:
5x^5 - 2x^4 - 25x^3 + 10x^2 + 20x - 8

To start, let's factor out the common factor from the polynomial, which is x-1:
(x-1)(5x^4 + 3x^3 - 22x^2 - 12x + 8)

Now, let's find the roots of the quadratic equation after dividing by (x-1). We use synthetic division to find that x=1 is a root. Factoring again:
(x-1)(x+1)(5x^3 - 2x^2 - 24x + 8)

Now, we need to find the roots of the cubic polynomial. Using synthetic division, we find that x=-1 is a root. Factoring the cubic polynomial:
(x-1)(x+1)(x+2)(5x^2 - 12x + 4)

Now, let's find the roots of the quadratic polynomial. By using the quadratic formula, we find the roots to be x = (6 ± √26)/5.

Therefore, the roots of the original polynomial are x = 1, -1, -2, (6 + √26)/5, and (6 - √26)/5.

Now, to determine the sign of the polynomial in each interval formed by these roots, we can evaluate the polynomial at a point in each interval.

For x < -2, pick x=-3 for example:
5(-3)^5 - 2(-3)^4 - 25(-3)^3 + 10(-3)^2 + 20(-3) - 8 = 3245 > 0

For -2 < x < -1, pick x=-1.5 for example:
5(-1.5)^5 - 2(-1.5)^4 - 25(-1.5)^3 + 10(-1.5)^2 + 20(-1.5) - 8 = -37.5 < 0

For -1 < x < 1, pick x=0 for example:
5(0)^5 - 2(0)^4 - 25(0)^3 + 10(0)^2 + 20(0) - 8 = -8 < 0

For 1 < x < (6 - √26)/5 and (6 + √26)/5 < x, the polynomial will be greater than 0.

Therefore, the solution to the inequality 5x^5 - 2x^4 - 25x^3 + 10x^2 + 20x - 8 <= 0 is:
-2 < x < -1