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
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