Question

(x^4 +1)(x^2 +2x-15)>0

Answer 1

(x⁴ + 1)(x² + 2x - 15) > 0

Заметим, что x⁴ + 1 > 0 при любых x, значит, первая скобка на знак влиять не будет и её можно отбросить:

x² + 2x - 15 > 0

x² + 2x + 1 - 16 > 0

(x + 1)² - 4² > 0

(x + 1 - 4)(x + 1 + 4) > 0

(x - 3)(x + 5) > 0

     +                           -                                  +

--------------(-5)-------------------------(3)---------------------------> x

\\\\\\\\\\\\\\\                                    \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Ответ: x ∈ (-∞; -5) U (3; +∞).

Answer 2

We can factor the left-hand side of the inequality as follows:

(x^4 + 1)(x^2 + 2x - 15) = (x^2 - 3)(x^2 + 4)(x + 5)(x - 3) > 0

To determine the sign of the expression, we need to examine the sign of each factor separately.

(x^2 - 3) is negative for x in the range -√3 < x < √3, and positive for x outside this range.

(x^2 + 4) is always positive.

(x + 5) is positive for x > -5 and negative for x < -5.

(x - 3) is negative for x < 3 and positive for x > 3.

To determine the sign of the entire expression, we need to consider all the possible combinations of signs for the factors.

Case 1: (x^2 - 3) < 0 and (x^2 + 4) > 0

This case is not possible because the product of two factors with opposite signs is negative, and we are looking for a positive expression.

Case 2: (x^2 - 3) > 0, (x^2 + 4) > 0, (x + 5) > 0, and (x - 3) > 0

This occurs when x > 3, in which case all the factors are positive and the expression is positive.

Case 3: (x^2 - 3) < 0, (x^2 + 4) > 0, (x + 5) > 0, and (x - 3) < 0

This occurs when -5 < x < -3, in which case two of the factors are negative and two are positive, resulting in a negative expression.

Case 4: (x^2 - 3) < 0, (x^2 + 4) > 0, (x + 5) < 0, and (x - 3) > 0

This occurs when -3 < x < -5, in which case two of the factors are positive and two are negative, resulting in a negative expression.

Case 5: (x^2 - 3) < 0, (x^2 + 4) > 0, (x + 5) < 0, and (x - 3) < 0

This occurs when x < -5, in which case three of the factors are negative and one is positive, resulting in a positive expression.

Therefore, the solution set to the inequality is x < -5 or -3 < x < 3, or x > 3.